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THE DYNAMICS OF 
MECHANICAL FLIGHT 



THE DYNAMICS OF 

MECHANICAL FLIGHT 

Lectures delivered at the Imperial College of Science 
and Technology, March, 1910 and 191 1 



BY 

SIR G. GREENHILL 



NEW YORK 

D. VAN NOSTRAND CO. 

TWENTY-FIVE PARK PLACE 

1912 



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CONTENTS 



Page 

Introduction 1 



I. General Principles of Flight, Light and Drift . 8 

II. Calculation of Thrust and Centre of Pressure 

of an Aeroplane . . . . . . 28 

III. Helmholtz-Kirchhoff Theory of a Discontinuous 

Stream Line ....... 48 

IV. Gyroscopic Action, and General Dynamical 

Principles 73 

V. The Screw Propeller ...... 89 

VI. Pneumatical Principles of an Air Ship . . 107 



M.F. 



THE DYNAMICS OF 

MECHANICAL FLIGHT 



INTRODUCTION 



The lectures were delivered in the Imperial College of Science 
and Technology in March 1910 and 1911, under the title 
" The Dynamics of Mechanical Flight," and they are given 
here in the form in which they were delivered. 

The subject was then beginning to take hold of the public 
imagination, consequent on Bleriot's feat of crossing the Channel 
on July 25, 1909, and the great strides made in the interval 
since in Mechanical Flight. 

The possibility of Human Flight has been an obsession of the 
imagination of Man from the earliest times recorded, for which 
an extensive article should be consulted in the Denkschrift der 
I.L.A. (der Internationalen Luftschiffahrt Austellung) Frank- 
furt, 1910, Band I, p. 118, Flugprobleme in Mythus Sage and 
Dichtung. 

In the Greek Mythology, Demeter rides in a car drawn by 
flying dragons, and Homer describes the flight of Hera in her 
chariot, Iliad V., 750; and then there is the legend of Icarus and 
his father Daedalus " who taught his son the office of a fowl, and 
yet for all his wings the fool was drowned " ; and another legend 
of Archytas of Tarentum, " aerias tentasse domos " with his 
invention of a flying mechanical bird. 

./Eschylus in his Prometheus has described the arrival in a 
flying chariot of the chorus of the Ocean Nymphs, followed by 
their father Oceanus on a four-legged bird, anxious to return in 
a single flight from the Scythian desert and the Caucasus to 
beyond the Pillars of Hercules and over the Atlantic. 

M.F. B 



2 INTRODUCTION 

The fabulous Life of Alexander the Great, by the Pseudo- 
Callisthenes, was a favourite book of the Middle Ages. 
Alexander is described here to have placed a yoke on the neck of 
two strong eagles that had been kept fasting for three days. When 
Alexander took his seat on the yoke, the eagles flew up with him 
in the air, wherever he pointed his spear, because the head of it 
carried a large lump of liver. This flying machine of Alexander 
is illustrated in manuscripts with four or eight eagles or griffins, 
shown in the vignette on p. 7, and representations in sculpture 
are to be seen in St. Mark's, Venice, and the cathedral of Basle. 

The Tartar and Chinese legend of the Bronze Horse is repro- 
duced in the Squires Tale of Chaucer — 

"Him who left half told 
The story of Cambuscan bold, 
And of the wondrous horse of brass, 
On which the Tartar King did ride " : 

known to us more recently in the operatic version of the Clieval 
de bronze of Scribe and Auber ; parodied also by Cervantes in Don 
Quixote II. in the description of Clavileno. 

Chaucer goes into detail of the Magic Steed, but he did not 
realise the difficulty of the mechanical problem in his jaunty 
description — 

" This same stede shall bere you ever-more 
With-outen harm, til ye be ther yow leste, 
Though that ye slepen on his bak or reste, 
And turn ayeyn, with wry thing of a pin. 

But whan yow list to ryden any- where. 
Ye moten trille a pin, stant in his ere — 
Bid him descend, and trille another pin. 

Trille this pin, and he wol vanishe anon. 

He that it wroughte coude ful many a gin/' 

So Hecate, too, in Macbeth, reaching for her stick, a broomstick, 
and saying, "I am for the air"; like Abaris on his arrow, 
putting a girdle round about the Earth between meals. 

But Dr. Johnson's "Rasselas," Chapter YL, "A Dissertation 
on the Art of Flying," anticipates very accurately the gliding ex- 
periments of Lilienthal and Pilcher, now resumed by the Wright 
brothers. The description is so curiously apposite that an 
extract may well find a place here : — 



INTRODUCTION 3 

Among the artists that had been allured into the Happy Valley, 
to labour for the accommodation and pleasure of its inhabitants, was 
a man eminent for his knowledge of the mechanic powers, who 
had contrived many engines, both for use and recreation. 

This artist was sometimes visited by Easselas, who was pleased 
with eve^ kind of knowledge, imagining that the time would come 
when all his acquisitions would be of use to him in the open world. 

He came one day in his usual manner, and found the master busy 
in building a sailing chariot. He saw that the design was practicable 
for a level surface, and with expressions of great esteem solicited its 
completion. 

" Sir," said the master "you have seen but a small part of what the 
mechanic arts can perform. I have long been of opinion that instead 
of the tardy conveyance of ships and chariots, man might use the 
swifter migration of wings, that the fields of air are open to knowledge, 
and that only ignorance and idleness need crawl on the ground." 

" The labour of rising from the ground will be great " said the 
artist " as we see in the heavier domestic fowls ; but as we mount 
higher the earth's attraction and the body's gravity will be gradually 
diminished, till we arrive at a region where man will float in the air 
without any tendency to fall ; no care will then be necessary but to 
move forward, which the gentlest impulse will effect." 

" Nothing," replied the artist, " will ever be attempted if all possible 
objections must first be overcome. If you will favour my project I 
will try the first flight at my own hazard. I have considered the 
structure of all volant animals and find the folding continuity of the 
bat's wing most easily accommodated to the human form. 

" Upon this model I will begin my task to-morrow, and in a year 
expect to tower into the air beyond the malice and pursuit of man." 

The Prince visited the work from time to time, observed its 
progress, and remarked many ingenious contrivances to facilitate 
motion and unite levity with strength. 

The artist was every day more certain that he should leave 
vultures and eagles behind him, and the contagion seized upon the 
Prince. In a year the wings were finished, and on a morning 
appointed the maker appeared furnished for Might, on a little 
promontory ; he waved his pinions awhile to gather air, then leaped 
from his stand, and in an instant dropped into the lake. 

His wings, which were of no use in the air, sustained him in the 
water, and the Prince drew him to land, half dead with terror and 
vexation. 

B 2 



4 INTRODUCTION 

These extracts from Easselas will show that Dr. Johnson had 
realised with more accuracy than Cyrano de Bergerac the 
difficulty of the problem to be solved. 

Herr von Lilienthal's experiments, followed by Prof. Fitz- 
gerald and Mr. Pilcher, have borne out to some extent his 
prophecy, which we hope will not come true in the gliding 
experiments of the Wright brothers, now in progress. 

But viewed by the cold, calculating eye of mechanical science, 
the usual poetical description is seen to be hopelessly absurd and 
impossible ; especially since Mr. Maxim proved to demonstration 
with his machine the enormous power required, out of all pro- 
portion to the size, for man to emulate the bird. 

How came the obsession to take so firm a hold on the human 
imagination as to the possibility of flight by a man like a bird ? 
Here is a subject worthy of careful historical examination. 

The pterodactyl of palaeontology goes to prove that flight was 
practised by an animal much larger than now is capable, judging 
also by his degenerate descendant archasopteryx. 

A mere decrease of gravity, due to the Earth not having 
shrunk so much, would not account for it, as the air density 
would have diminished in proportion. 

Flight would be no easier if gravity was halved, as the density 
of the air would be halved at the same time. 

We are led to conjecture that the atmosphere was formerly 
much denser than now, and that a large portion has leaked away 
and escaped into space, just as free hydrogen will now escape ; 
and that the supply has not been renewed by the comets, as 
imagined by Newton in the " Principia," Lib. III. : — 

" Porro suspicor Spiritum ilium, qui Aeris nostri pars minima est, sed subtilissima 
et optima, et ad rerum omnium vitam requiritur, ex Cometis prascipue venire."' 

The cooling of the Earth's surface and the deposition of the 
moisture in the air would not account for a diminution of atmo- 
spheric density, as aqueous vapour is only about half as dense 
as air ; but the carbonic acid may have been more abundant, 
before it was absorbed by a dense vegetation. 

A writer, Mr. Harle, in the American journal "Cosmos," on 
Atmospheric Pressure in past Geological Ages, has come to the 



INTRODUCTION 5 

conclusion that air density must formerly have been very much 
greater, judging from the pterodactyl with wing span over 
thirty feet, equal to a Bleriot machine, and fossil dragon-flies 
of the carboniferous era, over three feet between the tip of 
the wings. 

The winged figures of Assyrian art may prove an inherited 
reminiscence of the time of the pterodactyl, in an atmosphere of 
perhaps two-fold density. 

The figures have come down to us through the Greek 
Mythology to the representation by the modern artist of the 
winged angel, described by Milton in " Paradise Lost," and 
painted in the cathedral altar-piece. 

But while the poet and artist invites us to consider his human 
figure, Isis, Nike, Michael, as life size with wings proportioned 
to suit the aesthetic sense, mechanical science of similitude 
requires our imagination to reduce the scale to the size of the 
present dragon-fly, if flight is to be maintained on unassisted 
one-man power, in the tenuity of our present atmosphere. 

Pope, however, has grasped the principle of Mechanical Simili- 
tude in his " Bape of the Lock," where he arms his Gnome and 
Ariel Sprite with a bodkin spear : 

" Know, then, unnumbered Gnomes around thee fly, 
The light Militia of the lower sky. 1 ' 

A thought echoed by Tennyson's " Airy navies of the blue." 

In all the recent rapid progress there is one difficulty still 
unsolved ; a name has not been discovered for the flying machine 
of universal acceptance ; and here the poverty comes out of -the 
English language in the formation of a new compound word. 

The German calls it a dragon, as in the car of Demeter ; and 
Demoiselle (dragon-fly) is the name given by Santos Dumont to 
his mount. 

The lady of the " Bape of the Lock," reminded by — 

"Think what an Equipage thou hast in Air, 
And view with scorn two Pages and a Chair." 

might say then, " Bring the dragon-fly round to the door," or 
dragon instead of the brougham. 



6 INTRODUCTION 

Santos Duniont takes the dragon-fly as his mechanical model ; 
but the two inches of length of the insect becomes twenty-four 
feet in the Demoiselle, a linear scale of 144 to 1, and the speed 
must be raised twelve-fold, to over sixty miles an hour, from five 
or six in the dragon-fly. 

Size and weight mount up, then, 144 times 144 times 144 times, 
three million -fold, and a few grains in the fly become nearly half 
a ton in the Demoiselle. And the power required is another 
twelve times more, so we have to multiply by twelve seven times 
over, thirty-six million times, to go from the horse power of the 
fly to the dragon. 

This is the simple calculation which the mathematician has 
brought to the attention of the mechanical enthusiast, and 
so the mathematician, as Clifford or Herbert Spencer, 
is accused of proving that Human Flight was an impossi- 
bility. 

And so it was, as a retrospect of the last few years will show. 
It was impossible till the motor was available, strong and light 
enough to lift itself and three- or four-fold weight, including a 
man pilot. For the development of the motor we have to 
thank the Associated Motor Car industry, but for which we 
should still be waiting. 

Such a mechanical genius as Maxim, working about fifteen 
years ago, and with unlimited resources, was not able to achieve 
a flight, and so we may be sure no one else could. A few poor 
aeronautical enthusiasts might still be working, and for a hundred 
years more, without arriving at a result presented to them now 
ready made. 

All such experimenters, however, were doing valuable work in 
preparing the way for success, Langley for instance, subsidised 
by the American Government, so that the moment the motor was 
available, man could fly. 

But not with one man power, as in the artist's picture. The 
pilot likes to ride on 50 horse-power at least, and the power is 
mounting up to 100 horse-power and more, before he will secure 
the speed to become the real mechanical enlargement of the 
model dragon-fly. 

Roger Bacon has described the direction he would like to 



INTRODUCTION 7 

follow in his inventions, impossible to carry out in his own 
day from lack of mechanical skill. 

Beginning with the Motor Car, he would follow on with the 
Flying Machine, and then the Diving Dress, and the Suspension 
Bridge perhaps. But Bacon is as cautious as Herodotus in 
distinguishing between what he has heard only, and seen him- 
self, as we read in these extracts — 

De secretis operibus artis et naturae, Caput LV. 

De instruments artificiosis mirabilibus. 

— Currus etiam possent fieri ut sine animali moveantur cum impetu inasstima- 
bili, ut existimantur currus falcati fuisse quibus antiquitus pugnabatur. 

Possunt etiam fieri instrumenta volandi, et homo sedens in medio instrumenti 
revolvens aliquod ingenium, per quod aire artificialiter composite aerem verberent, 
ad modum avis volantis. 

Possunt etiam fieri instrumenta ambulandi in mari et in fluviis ad fundum sine 
periculo corporali. Xam Alexander magnus his usus est, ut secreta maris videret, 
secundum quod Ethicus narrat astronomus. 

Hsec autem facta sunt antiquitus, et nostris temporibus. 

Et certum est, prater instrumentum volandi quod non vidi nee hominem qui 
vidisset cognoyi, sed sapientem qui hoc artificium excogitavit explicite cognosco. 




(British Museum M.S.) 



LECTURE I 

GENERAL PRINCIPLES OF FLIGHT. LIGHT AND DRIFT 

A simple preliminary calculation will show the requirements 
indispensable for a flight in the air of an aeroplane machine. 

Take an aeroplane, AA', rectangular, moving horizontally 
at a slope a in still air with velocity Q f/s (feet per second) 
(Figs. 1, 2, 4). 

This is the velocity as observed by a spectator standing on 
the ground. 

But the pilot on his seat, looking ahead, is unconscious of his 
own motion, and feels the air blowing past with velocity Q f/s; 
and the dynamical problem is the same from each point of 
view : of the pilot who thinks himself stationary, as a Wright 
glider, and the air blowing past, or of the spectator standing 
in still air, and watching the pilot flying past with velocity Q f/s. 

But when the spectator is standing in a wind blowing over 
the ground with velocity W f s, he will notice a difference in 
the speed of the machine, Q + W f/s, according as it is flying 
up the wind or down. 

The pilot, however, will not be aware of a difference any more 
than a bird in a gale, as his machine cleaves the air with the 
same speed as before. 

Suppose a bird then, or an aviator is to make a flight in a 
wind ; he will start for preference against the wind, so as to 
increase his relative velocity, and so rise quicker. It is easier 
then, to rise against the wind than with the wind behind ; and 
to alight also, provided the pilot does not descend into a region 
of relative calm. 

The aeroplane A A' is now supposed up in the air, with the 
wind blowing past it at Q f/s; and a pressure difference arises 
in consequence, which gives a resultant thrust, T lb, on the 
under side of the plane. 



GENERAL PRINCIPLES OF FLIGHT 9 

The vertical component. T cos a, is called the Lift, as it is 
required to lift the weight. W lb ; and T sin a, the horizontal 







Fig. 1. 




component, is called the Drift, and this drift has to be overcome 
by the thrust of the screw, working at 

a) 



I sin a ^—. effective horse power (E.H.P.) 
ooCT 



reckoning 1 H.P. at 550 ft-lb sec. that is, 33.000 ft-lb min 
on Watt's estimate. The ratio then of 
Q 



(2) E.H.P. 



T sin a - 



500 Q tan a 



lift 



500 n 



or 



lift ~ Tcosa " 550 ~ 550 n' w E.H.P. Q 

if the slope of the aeroplane is reckoned at 1 in n ; but 
it is prudent to double the E.H.P. to arrive at an estimate 
of the I. H.P. ("indicated horse power), reckoning engine and 
propeller efficiency at 50 per cent. 



10 THE DYNAMICS OF MECHANICAL FLIGHT 

When the speed is given by S in m/h (miles an hour) the 
factor 550 is replaced by 375, which is 33,000 -r- 88, as 1 mile 
an hour is 88 feet per minute. 




Fig. 



I 



If the weight to be lifted is m times the weight of the motor, 
the motor must weigh 



(3) 



550 w 

Q m 



lb/E.H.P. 



to be halved with 50 per cent, efficiency to 

(4) 550n lb/LH.P. 

w 2Qm ' 

so that at S = 40 m/h, say Q = 60 f/s, and with n = 5, m = 2, 

as in Maxim's machine, the motor must weigh 

550 X 5 2,750 _,. „ /TTTT) 

2 X 60x2 =-24(r' Sa y 11 ' lb / LRR ' 

the figure given by Maxim ; and Herbert Spencer and Clifford 



GENEEAL PRINCIPLES OF FLIGHT 



11 



could tell the aeronaut with certainty that his ideal must 
await the motor which could be brought down to this weight, 
11 lb/LH.P., and lower still, before his machine could fly. 

The petrol motor had not come to any perfection in the day 
of Maxim's experiment, 1895, and so he was obliged to carry a 
boiler up with his engine. This implied a great increase in size 
all round, so that Maxim's machine (Fig. 3) weighed close on 
four tons, quite unmanageable in the light of modern experience. 

An aviator is akin to the imperfect orator, as his chief 
difficulty arises when he wants to get down ; and the shock of 
landing of four tons rather abruptly would be terrifying, with 
a boiler close by full of steam at 275 lb/inch 2 . 




Fig. 3. 



A modern locomotive will weigh complete about 80 tons, and 
will indicate 1,000 H.P. at 50 to 60 m/h ; the 

9 970 

H.P./ton is then 12-5, and the lb/H.P. = ^f = 180, 

so that no locomotive can be expected to fly, still less a marine 
engine, with its additional weight of condenser. 

This simple arithmetical calculation, of Herbert Spencer and 
Clifford, it can hardly be called mathematical, is enough to show 
why mankind was compelled to await the advent of the light 
petrol motor; and man might still be waiting, but for the 
previous development of the motor car. 

A list is given here of the chief dimensions of the early 
p ioneering machines ; the list can be added to, and brought up 
to date with each successive pattern and development. 



2 THE DYNAMICS OF MECHANICAL FLIGHT 





Maxim. 


Weight. 


Fabmas. 




(Nature, 
Aug. 1, 1895.) 


(Details missing.) 


(Flight. 
Oct. 10. 1909.) 


Speed S (m/h) ... 


40 


— 


45 


Q (f/B) 


60 


— 


66 


Wing Area A (ft 2 ) 


4000 


— 


410 (biplane) 


Slope 1 in n = 


4 to 5 


— 


— 


Lift or Weight, lb, 


10,000—8,000 


— 


1,212 + pilot 


Drift, lb, 


•2,000 


— 


— 


I.H.P. 

Lift " 


S^-0-048 


— 


— 


LH.P 


384 


— 


— 


Weight of Motor . . . 


4,000 


— 


— 


in4b/H.R 


11 


— 


— 



GENERAL PRINCIPLES OF FLIGHT 



13 



VOISIN. 

{Flight, 
Aug. 14, 1909.) 


Antoinette. 

{Flight, 
Oct. 20, 1909.) 


Bleeiot. 

{Flight, 
July 31, 1909.) 


Santos-Dumont 

Demoiselle. 

{Flight, 

Oct. 2 and 9, 1909.) 


450 (biplane) 


365 


200 


115 


1,150 


1,040 + pilot 


700 


400 


230 


— 


140 


80 


55 


— 


34 


20 


400 


— 


200 


110 


8 


— 


6 


5-5 



14 THE DYNAMICS OF MECHANICAL FLIGHT 

But in the machine of the present day the H.P. of the motor 
has risen from 75 to 150, and the weight to over 500 kg, 
or half a ton. 

The diagrams, Fig. 1 of a monoplane, and Fig. 2 of a biplane, 
are based on figures in the aeronautical journal Flight, as well 
as the numerical data above. 

The year 1909 may be taken as the epoch when the dreams of 
fancy were realised which have inspired the artist and poet of 
romance as far back in history as we can trace, through Tennyson, 
Chaucer, up to Homer and his unknown predecessor, Egyptian, 
Babylonian, and Chaldaean. 

But now anxiety is past for the motor, we can calculate 
dimension and design for a given weight or lift at a speed 
assigned ; and the first mathematical requirement is a deter- 
mination of the thrust T lb, over an area A ft 2 , driven through 
the air with velocity Q f/s, at a slope a or 1 in n. 

An exact mathematical treatment brings in the Schwarz- 
Christoffel theory of conformal representation (mapping), and 
its first application by Helmoltz and Kirchhoff to the stream lines 
of liquid through an orifice or past a barrier, and of the 
discontinuity arising at an edge. 

This theory will represent the substance of these Lectures ; 
but we may lead up to it through the ancient assumption of 
Newton, which treats the medium as devoid of rigidity ; in the 
language of the " Principia," p. 219, second edition ; 

" In Mediis quae rigore omni vacant resistantire corporum sunt in duplicata 
ration e velocitatum." 

The medium of air is taken then to behave like a cloud of 
particle dust, as assumed also in the modern kinetic theory of 
a gas. 

Taking C ft 3 of the medium to weigh 1 lb, we say that the 
specific volume (S.Y.) is C ft 3 /lb. 

Thus, for air in a room, 12 to 13. ft 3 weigh 1 lb ; an 
average S.V. of 12'5 ft 3 / lb, or 28,000 ft 3 / ton. 

A room 30 feet cube will then contain nearly a ton of air. 



GENERAL PRINCIPLES OF FLIGHT 15 

The density of air would thus be 0*08 lb/ft 3 , but, as usual in 
all calculation, we prefer to work with the integers of the 
reciprocal. 

If the plane is A ft 2 in area, and sloped at an angle a, the 
vertical aspect in front elevation is A sin a ft ; and the air 
received on the plane is 

(5) QA sin a ft 3 /sec, or z— sin a, lb/sec. 

Taking the air particles as inelastic, they slide along the plane 
after impact without interference ; their momentum perpen- 
dicular to the plane is reduced to zero ; so that in t seconds, 
the thrust T lb will generate or destroy Tt sec/lb of momentum 

in reducing ^ — t sin a lb from velocity Q sin a to ; and 

then 

rc\ rr± QA . • Q sin a m A Q 2 . 9 n 

(b) It = -£_ t sin a , I = — — sin- a, lb. 

C g C g 

The gravitation measure of force and momentum is employed. 
as in all engineering calculation. 

The theory is exact when the direction of the leaving stream 
is compelled to be parallel to the plane, as would be the case 
for a horizontal jet of vertical cross-section B = .4 sin a ft, 
if it was received on a plane barrier of unlimited extent, and then 

(7) T = - 2 sin a, lb. 

c 9 

The theory can be employed in all calculations of hydraulic 
machinery, as a turbine or Pelton wheel (and a screw propeller), 
where the water is compelled to follow a definite path. 

Later on an examination will be made of the action of a screw 
propeller, and the various theories that have been advanced ; 
and the theory which seems to hold the ground with the naval 
architect is the one given by Rankine in the " Transactions of 
the Institution of Naval Architects, 1865," in which it appears 
that Rankine's formula is a direct result of the Newtonian theory 
and assumption just stated. 



16 THE DYNAMICS OF MECHANICAL FLIGHT 

In air the particles may be supposed elastic, endowed with a 
coefficient of restitution e ; and this makes 

(8) T = (1 + e) 4 9l sin 2 a? lb 

C g 

or 

(9) | = (1 + e) g sin 2 a, lb/ft.* 

On this assumption for an aeroplane, the lift or weight which 
can be lifted is 

(10) W = T cos a = (1 + e) ^-? 2 sin 2 a cos a, lb, 
at a rate 

(11) ? = (1 + e) ^ sin 2 a cos a, lb/ft 2 , 

and to overcome the drift T sin a, the 

(12) H.P./ft 2 =Jsma| = (l + ,) g§ |^smS a . 

In a numerical calculation it is convenient to take, in round 
numbers, 

(13) g = 32 f/s 2 , with C = 12-5 ft s /lb, 

for air, as this makes 

(14) Cg = 400, 
so that we can write 

(i5, §=(!+*) ( < 4ipy> ib / ft - 2 

(16) H.P./W = (1 + e)(*™±)' %£-« -*>(! + ^ 



20 y 550 550 v 7 V 20 

With normal impact a = 90°, and even with e = 0, this 
gives 

a wind of 20 f/s giving a pressure of 1 lb/ft 2 ; or if the velocity 



GENEEAL PKINCIPLES OF FLIGHT 17 

Q f/s is the equivalent of S m/h, 

Normal atmosphere pressure is reckoned at 15 lb/inch 2 , which 
is 2,160 lb/ft 2 , nearly 1 ton/ft. 2 

But even at a speed of S = 100 m/h, the normal pressure 
in (19) is 54 lb/ft 2 , which is only gfeth of an atmosphere, or 
| inch in the mercury barometer, so that air compression may be 
ignored, and the density taken as uniform, until we come to the 
high velocity usual in the screw propeller. 

These numbers, even with e = 0, are greatly in excess of the 
result of recent experiment and theory ; although at the Forth 
Bridge, in a gale of 78 m/h, a pressure of 70 lb/ft 2 was recorded ; 
but this abnormal result may be attributed to gustiness. 

To drive the plane normally would then require 



On the other hand, with the thrust varying as the square 
of sin a, the formula gives numbers much too low at a small 
slope of the plane. 

Apply the formulas to Maxim's machine, where the area 
A — 4,000 ft 2 , at a slope of 1 in 5, so that 

sin a or tan a = — ' and cos a = 0'98 may be replaced by unity. 

At a speed of Q = 100 f/s the lift would be 4,000 lb, only 
half the weight to be lifted, so that the wing area should be 

double, 8,000 ft 2 ; and the E.H.P. would be ^^ X ^, say 

5 550 

150, requiring nearly the full I.H.P. of 360, with a propulsive 

efficiency of 40 per cent. ; this would prove that the machine 

cannot leave the ground at this speed. 

m.f. c 



18 THE DYNAMICS OF MECHANICAL FLIGHT 

But the modern mathematical theory of Kirchhoff, to which 
we return later, confirmed by Langley's experiments, proves that 
the lift is given more nearly by 

(21) A (^ty I tt sin a, instead of A (^j 2 s in* a, 

so that for a small angle a the lift varies more nearly as sin a 
than ( sin a ) 2 , much greater than given by the ancient theory. 

Applying this again to Maxim's machine at a speed Q = 100 f/s, 
the lift 

(22) T = 4,000 (~°Y x 0-7854 x 0-2 = 16,0001b, 

double the weight to be lifted ; and if the speed is reduced to 
#=45 m/h, Q = 66 f/s, the lift is halved, and the machine could 
rise at this speed, with the engines working up to full 360 H.P. 

But at Baldwyn's Park, the run of about a quarter of a mile 
was not long enough to get up this speed. 



Fig. 4. 



For normal incidence on Kirchhoff's theory the factor 



( 23) rfr,-^'-T 

is required to reduce Newton's coefficient ; this makes the 
formula for wind pressure 



GENERAL PRINCIPLES OF FLIGHT 19 

(24) 0-U x 54 (J^f - 21 (^ lb/ft ? 

mach lower than 32 (t™) , as given by Stanton and Eiffel. 

The wide divergence of the Xewton formula is explained on the 
Kirchhoff theory by the shape of the stream lines, AJ and A' J', 
as they leave an edge, A and A', of the barrier AA'. These are 
not straight and in the line AA' , as in Xewton's theory, but 
curved backward, as in Fig. 4, forming a skin or bounding 
surface between the wind and the dead air behind the barrier. 

(Experiment with a knife -blade held in the jet of water from 
a tap.) 

We proceed now to the object advertised in these Lectures, the 
determination of the Stream Lines past a Plane Barrier, and 
of the Discontinuity arising at an Edge (Report 19, Advisory 
Committee for Aeronautics, TVynian, Fetter Lane, E.G., 1910) 
as it is determined by recent mathematical research ; and a 
digression must be made on the history of the subject inaugurated 
by Helmholtz and Kirchhoff, about 1868. 

Although Helmholtz was the pioneer of the theory, in 1868, 
we begin with the result arrived at by Kirchhoff, about a year 
after, in 1869, as it gives a solution for an aeroplane A A' 
(Fig. 4.) 

The stream of air, coming from infinity at I with relative 
velocity Q, is split along a curved stream line IB, and divides at 
B on A A' into the two stream lines BAJ and BA'J', where A J 
and A' J' are curved stream lines proceeding from the edge A 
and .4'. 

Along A J and A' J' the velocity must be constant and equal 
to Q, so as to ensure that the pressure in the dead air behind 
A A' is the constant atmospheric pressure; and the analytical 
difficulty is to secure this condition, a difficulty which Kirchhoff 
was the first to overcome. 

c 2 



'20 THE DYNAMICS OF MECHANICAL FLIGHT 

Kirchhoffs result states that the motion of the medium, treated 
as homogeneous, is given by 



(25) ch O = cos a + 



J(iv B - IV) 



where a denotes the angle the stream at IJJ' makes with the 
plane AA', and A T is a constant. 

In the notation of this subject 

(26) z = x + yi, w = (f> + \j/i, 

4> denoting the velocity function at the points (x, y) such that 

the velocity q is given by the downward gradient j- of the 

function 4> , and \j/ is the conjugate stream function, constant 
along a stream line. 

Also, if the velocity q at (x,y) makes an angle 6 with Ox, 

(27) *t = ^=-gco S ^=-^ = - 9 sin0; 
dx dy ay dx 

and then 

/oo\ dw dd> . ■ ddr /),••/> 

The function (Tand & is now introduced, defined by 

(29) ° = t=-Q c l± =9 (cos 6 + i sin 6) = 2 e 6l ' 

v ' e> diu q q 

(30) O = log 9. + 01. 



(31) ch Q = ch log — cos + * sh log — sin 

to be employed in equation (25). 



GENERAL PRINCIPLES OF FLIGHT 21 

Along the dividing stream line we take 

(32) if/ = 0, w B — w = <f> B — <j>, 

and from I to B 

(33) <x > <£ > cf> B , <j> B - <f> is negative, \J($~$) imaginary, 

so that 

(34) ch log — cos 6 = cos a, sh log — sin 6 = , 



q q VW-Qb) 

But beyond B 

(35) 4> B > $ > - oo, y(^B — </>) and v'fe - w ) is real 

sh log -^ sin = 0, 

so that either 

(36) sin 6 = 0, = along 5.4, = tt along BJ.'; 
or else 

<37) sh log 5 = 0, 9.= l,q=Q, 

9. <1 

as required along the stream line A J and A' J' ; and then 

N 

(38) ch log — = 1, cos 6 — cos a 



9 J(<f>B ~ <M 

Along the stream line .4 J" 

(39) ** = - 0, ^ - * = g s , 

if the arc AP = s is measured from .4, and 

(40) QS = 4>A ~ $B + <t>B - * = +A - 4>B + 



(COS 6 — COS a) 

= nT. JL J— ] 

L(C0S — COS a)' 2 (1 — COS a) 2 J 

the intrinsic equation of the stream A' J' with 0<#<a; 
with a similar expression for A 'J'. 



22 THE DYNAMICS OF MECHANICAL FLIGHT 

For normal incidence (Fig. 5) a =90°, cos a = 0, and the 
curve A J is the evolute of a catenary, given by the intrinsic 
equation 

Qs = N 2 tan 2 0, s — c tan 2 0. 



The theory requires the existence of the counter current BA T 
{nappe dor sale in French), passing over the attacking front edge 
A' of the plane AA' ; this nappe dor sale is usually omitted in 
the diagram of popular explanation as insensible, but the 
existence is revealed in a photograph of a current, either of air 
or water. 



JWT^: 




i A 



Fig. 5. 
It will be proved in the sequel that 



(41) 



BA' 
AA' 



i(l — cos a) — \ sin a sin 2 q -j- I q sin a 



and at a small angle a, in radians, we may replace 

(42) a by sin a + -|-~sm 3 a, and cos a by 1 - \ sin 2 a — i sin 3 a, 

giving the approximation 

BA' 17 



(43) 



A A' 48 



sin* a. 



Thus at a slope of 1 in 5. 



44 



sin q = 0-2, 4J = 27000 - °-° 005 



GENERAL PRINCIPLES OF FLIGHT 23 

and even at 2 in 5, 

BA' 
(45) sin a = 0-4, -jjr = 0-008 ; 

so that the counter current would still be hardly perceptible. 
But with normal incidence (Fig. 5), 



BA' 
AA 



(46) a = J 7T, sin a = 1, -j-t- = J, 



and the current divides equally. 



Anticipating other results of KirchhofFs theory, to be proved 
hereafter, we find 

(47) — = — — ■ — : lb/ft 2 , versus ~r = ^r sm 2 a, 

v ' A Ug 1 + | 7r sm a ' ' ^4 C^ 

of the ancient theory ; also that the centre of pressure L is in 
front of F the centre of A A', between F and A', where 

sao\ FL -A- COS a -,-i-r n 

(48) -m = ., — ^ = > versus FL = 0. 

v 7 AA 1 + J 7r sin a 

The look of these formulas (47) and (48) suggests a geometrical 
representation, as in Fig. 6, associated with the ellipse, whose 
focal polar equation is 

7 2 7 

( 49 ) r = a + c cos = 1 + c cos 0' 

(50) cos = sin a, and eccentricity e = \ -k = 0*7854, or -^i. 

Q 2 
Take FX&s the unit to represent ~ r geometrically, the average 

pressure in lb/ft 2 on Newton's theory for normal incidence, 
a = ^ 77 ; and make 

(M) FL ° i ^ - I _ - 0-44 



24 THE DYNAMICS OF MECHANICAL FLIGHT 



Draw the ellipse APL , with focus F, vertex A, serni-latus 



rectum FL , and eccentricity 



Draw the semicircle FQX on 



the diameter FX, cutting FP in Q. Then 

T T 

A J 7r sin a LP A_ 

W = l + iirSilL«= ^X' Q*_ 

Cg Cg 



(52) 



sm- a 



LP' 

la: 



if QP' is drawn parallel to FL ; so that the curves L PA 
L P'X represent T and T' graphically. 
N 



L 
I 


\p\p 


/r 




A 








F 


>4 

Fig. 6. 


X 



Also 
(53) 



FL 
FL. 



FP 

FL. 



COS a 



COS a 



1 + 1 



giving L the centre of pressure when FL is ^ the breadth 
of the plane A A' ■; and L lies inside the middle f of AA'. 

This can be verified experimentally with a plane plate AA' in 
a current of water ; pivoted like a balanced rudder about an axis 
through L, and then measuring a. 

If L coincides with F, the plane sets itself at right angles 
to the current with a = \ it ; as L moves away from F, a dimin- 
ishes to zero when FL = T 3 g AA ' ; as L is placed still further 
away from F, the plane A A' still remains in the line of the 
stream. 



GENERAL PRINCIPLES OF FLIGHT 25 

The vane of a weathercock could thus be pivoted so as to point 
at any assigned angle with the w T ind ; pivoted at the centre F 
the vane would set itself across the wind. 

The movement of L towards the leading edge A' shows why a 
flat plate if free tends to set itself broadside to the relative stream 
as a position of stable equilibrium, seen realised in a falling card 
or leaf ; and it explains the instability of the axial motion of an 
elongated body like a ship. 

The stability of the course of a ship is secured only by a 
constant attention to the helm ; but in a flying machine, by 
increase of Aspect Ratio, making the spread of the wings much 
greater than the axial depth, the vertical rudder requires little 
attention, but the vertical stability of the course requires 
incessant control by the horizontal rudder. 

The curve L P for moderate value of a is seen running for 
some distance outside the curve L P f ; and this shows how on 
Kirchhoff's treatment the lift may be obtained with a wing area 
much smaller than was credited on the ancient theory, although 
a smaller pressure is assigned for normal impact. 

For instance, with an average n = 6, and at 45 m/h, 

<W lb- of lift _ 375 x 6 _ 

V ; thrust H.P. ~ 45 DU ' 

and on the ancient theory, 

/kkv wing area in ft 2 A 1 



Lift in lb T cos a M , x Q l . 

(1 + e) ^- sin a cos a 



eg 



practically 



ay 

for a small angle of 1 in n ; and with n = 6, Q = 66, 



" e > * 20 



, 56) Area 36 3_^_ 

Lift= (i+«)sr i+e ' 

varying from 3*3 to 1*65 with e from to 1. 



26 THE DYNAMICS OF MECHANICAL FLIGHT 

This is very much greater than is required in an actual 

400 115 

machine ; Bleriot takes =~~ ft 2 /lb, the Demoiselle -^ ft 2 /lb. 

But an inspection of the figure shows that at small value of a 
the lift given on the KirchhorT theory is much greater, requiring 
smaller wing area for given lift, 

,~„, Wing area 1 1 + I ir sin a 

lift " / Q \ 2 \ ir sin a cos a 



,20 
and this can be replaced by 



(58) — : — - — "t~ 4 ff , or practically - — 



n 

or 



QV i*--' r J 20/_g_y 42/a 

2oy V ioo y viooy 

This, for n = 6, S = 75, works out to 

7 
: ., K >2i or a little less than 1 ; 

42 (m) 

still much greater, nearly double, than is required in practice for 
wing area per lb of lift. 

The discrepancy is attributed to the gain in efficiency due to 
camber of the wing ; and a practical formula in general use is 

(59) ™ = i fiy, lb/ft*, 

Area n \ 8 / 
where n is the slope of the chord of the camber. 

We have shown that the counter current is insensible at the 
leading edge A' of attack ; and as 1 in n is understood as the 
slope of the chord of the camber, the slope at the rear edge A is 
about 2 in n, which reduces the ft 2 /lb of wing area to a close 
agreement with practice. 

But, as stated in the first clause of Report 19, there is no exact 
analytical theory at present for the calculation of a stream past 
a cambered wing, unless of two planes bent at an angle, and here 
the complication becomes almost intractable for practical use. 



GENEKAL PEINCIPLES OF FLIGHT. 27 

A liberal estimate of petrol and lubricating oil consumed is 
at 1 lb/H.P. hour, or one gallon per 10 H.P. hour : thus in a 
flight of 100 miles in two hours and a half at 40 m/h with a 
50 H.P. Gnome engine, the quantity required would be 

10. K 

50 x 2i = 125 lb, bulking-^- = 18-75, say 20 gallons, 



with a petrol of S.G. -f. 
With given lift, n varies as Q' 2 , making the 

H.P. Q 1_ 

lb ~ 550 n a Q' 

and the hours of a journey varying inversely as Q, the H.P. 
hours vary inversely as Q 2 ; so that about half the petrol is 
required to be carried if the speed over the journey is increased 
from 40 to 60. 

On the ancient theory, n 2 varies as Q 2 , making the H.P./lb 
constant, and the H.P. hours of a journey, and the petrol to be 
carried, inversely as the speed, or two-thirds for an increase of 
speed from 40 to 60 m/h. 

This calculation ignores friction and head resistance, but it 
indicates in a general way the advantage and economy of high 
speed in flight. 

Stability too is improved by speed, as well as economy in fueL 



LECTUEE II 

CALCULATION OF THRUST AND CENTRE OF PRESSURE OF 
AN AEROPLANE 

The first lecture was confined to generalities, and mathematical 
detail required for a complete study of the subject, was avoided 
as much as possible. 

But now we begin with a description of the Schwarz-Christoffel 
method of Conformal Bepresentation, or Mapping, which shows 
us how to re-invent the original Helmholz-Kirchhoff solutions, 
and to extend them with ease and certainty to a large number of 
similar problems of greater generality, for which Beport 19 may- 
be consulted, on the Stream Line past a Plane Barrier, 1910. 

The notation required has been given already in Lecture I., and 
Ave now proceed to state the theorem of Conformal Bepresentation 
in the essential form as required for the subsequent application. 

A point P, whose position is given by the vector z = x + 2/i, is 
to travel round a closed polygon or curve ; and it is required to 
represent z as a function of some variable u, such that 

(i.) u is real and diminishes from + oo to — <x> as P performs a 
circuit of the polygon, 

(ii.) points inside the polygon are to correspond to u = p -j- qi, 
but to u = p — qi for points outside. 

The relation will be given by 

fj z a 

(1) Wu = M (u - a)~i , 

and the product of factors of the same form ; and here a = a at a 
corner of the polygon, where the direction, or course, changes 
suddenly through a the exterior angle of the polygon. 



THRUST AXD CENTRE OF PRESSURE 



29 



The representation is called Conforcnal, because a small square 
on the z diagram corresponds to a small square on the u diagram, 
and the relative bearing of adjacent points is preserved for the 
two maps z and u , as they may be called ; and there is thus no 
distortion although there is rotation and change of scale. 

Conformal representation is then the same problem as 
mapping, and the first application on a large scale was the 
construction of a map. on a given system of projection. 

Thus for instance the relation 



(2) 



tan i- u 



connects the z stereographic projection of a hemisphere with the u 
Mercator chart. (Fig. 7.) 




Fig. 7. 

Suppose P travels round the polygon so that the inside is to 
the right hand (" starboard " a sailor would say), as in going round 
the clock; the angle a is positive when the change of course is to 
the right or starboard, due to porting the helm. 

If the helm is put to starboard the angle a would be negative. 

Considering a single factor of (1), the relation 



(3) 



dz 



M 



(■ - ■ ) 



where M may be complex, and we put 

(4) M = Ne e: = N (cos + i sin 



30 THE DYNAMICS OF MECHANICAL FLIGHT 

and this leads to 

,— = N [u — ft sii 



(5) : JV (m — ft) cos 0, ^~ 



rfw 



sin 



ft 7 ?/ 



— tan 0, so long as % > a, 



so that 

< 6 > Ac 

and P describes a straight line at an angle 6 with Ox. (Fig. 8.) 



A< 




But as m passes through a to the other side, where a >jm, and 
16— ft negative, 



<7: 



<8) 



dz id — EL — — 

^ = ^ e (-1) „(«-«) „ 

= Ne % e~ La (ft — 2i)~^ with — 1 = e*" 

= N [cos (0 - a) + i sin (0 - a)] (ft - u) 
,-- d >J at.:. 



^ = iV r COS (0 - a) (ft - tt) ~i , ^ = tf sin (0 -a) (ft - tt)" 



(9) ^ = tan(0-a), 

so that P describes a new straight line, at an angle — a with Ox 



THKUST AND CENTRE OF PRESSURE 31 

In the case of a single angle A, an integration gives 

i — — 
(u — a) * 



(10) 8-Z A = M 



1 - 1 



clz M 


z - 


u — a 


die ~ u — a' 


~ c & C — ft' 



The case of a = - requires special consideration ; here the 
interior angle of the polygon is zero, and the point A is at infinity ; 
the two sides of the polygon are parallel, shown meeting at A in 
the diagram in the conventional way of Euclid I. 27 (Fig 9) ; and 

(ii) 

If the two sides are parallel to Ox, M is real, and equal to B 
suppose ; and if h is the distance between them, we shall find 

h = TV B. 

The point P goes off to infinity at A along the lower line to 
the left, and returns from infinity towards the right along the 
upper line, as u diminishes through a ; and from (11) 

(12) x - x c +i(y.- y c ) = B log u ~^ 

C — CO 

x - x c + i (y - y e — h) = B log a _^Jl 



(13) 


u — 


a 


= e B 


1 cos 


y 


- Vc 
B 


+ i 


sin 


y - yc 


c — 


a 


B 








X — X, 




COS 


y - 


• y — 

B 


h 






= e B 


(- 


- i sin 


which 


requires 
















(ii) 






h 
cos-. 


= - 1, 


sin 


h 

B 


= o, 


h 
B 


= 77, 



y — y c - h 



so that the conformal relation for the space between two 
parallel lines in distance // apart is 

15 z - z e = - log 

77 c — a 



32 THE DYNAMICS OF MECHANICAL FLIGHT 

The point P now describes the lower line parallel to Ox from 
right to left, and goes off to infinity on the left hand at A, where 
u = a ; it turns the corner here through an angle a =77 to starboard 
and conies back along another parallel line at a distance h above, 
so as to have the interior of the space between the two parallel 
lines on the right hand. 

The same argument as in (3) applies to each of the corners, 
A, Ai, A 2 , ... of the polygon, where u = a, a±, a 2 , . . . , 
and to each of the factors in (1) so that 

/I2 a a i a - 

(16) — = M [u — a) V(u — a±)~ n u — a 2 ) ~ ir. . . , 

CLIO 

and no break occurs in the direction of motion except at the 
factor corresponding to the corner traversed of the polygon, so 
that all the factors may have their variation ignored at this point. 
The integration in (16) now requires special consideration, 
and is not always tractable, as we see in the sequel. 

In the applications to streaming motion we do not consider the 
z = x + yi polygon, as in the Electrical Applications given in 
J. J. Thomson's Eesearches ; but we draw the polygon for 

w = <£ + i/f i, and O = log — + i, 

and determine the conformal representation of each in terms of 
the same variable u by means of the appropriate expression for 

— and ^formed as in (1). 
du du 

Herein is the great improvement on the original method of 
Helmholtz and Kirchhoff, introduced by J. T. Michell, Phil. 
Trans., 1890, and improved by A. E. H. Love, Proc. Cam. Phil. 
Soc, 1891. 

Then if we have found w =/(w), & = F (u) by an integration, 
n dz ,dw F{ %, , N 

( 17 ) - Q cr l r^c^ e f {u) > 

and another integration will give z as a function of it ; but this 
integration need not be carried out, except when required for the 
determination of a geometrical length on the diagram. 



THRUST AND CENTRE OF PRESSURE 33 

The fluid is bounded by one or more stream lines, over which \j/ 
is constant, and represented by parallel lines in the w diagram, 
and the angle a in the w polygon is either -f it or — it, so that 

(18) — = product of factors of the form (u — a) or (u — b) 
du 

which can be resolved into a quotient and partial fractions 

of the form 

M 

u — a 

Now, considering the XI polygon, where 

(19) Q = log Q + Oi, 

the 12 polygon is composed either of parallel lines of constant 
and variable log—, corresponding to a boundary or barrier of z, 
or else a line at right angles of constant velocity q=Q, making 
log— = 0, while is variable, as over the free surface of a iet. 

Here is the advantage of X2 = log £ over ( = — e ° l , as con- 
stant q = Q would give an arc of a circle on the C diagram, 
and 6 would be constant along a radius ; and so the procedure of 
Helmholtz and Kirchhoff in employing Q is not the simplest, and 
was much improved by Planck's idea of using log C= &• 



Begin with the application to Kirchhoff' s problem, where a 
plane barrier A A' like an aeroplane is placed at an angle a 
across an infinite current of fluid, moving when undisturbed with 
velocity Q. 

In the disturbed motion a wake is formed in rear of AA\ 
which may be supposed still or turbulent, but at constant 
atmospheric pressure ; and the wake is bounded by the two 
free surfaces A J, A' J', extending to infinity at J and J', over 
which the pressure is constant and atmospheric. 

M.F. D 



34 THE DYNAMICS OF MECHANICAL FLIGHT 

The fluid is bounded by the single stream line J ABA' J', over 
which we take f = (Fig. 10). 

At the branch point B, where the stream divides, the velocity is 

zero, and^— = when u = b. 
on 

The w diagram consists of the single straight line \f/ = 0, but 
doubled back on itself at u = b, so that coming from infinity at j 
along the under side of the line with the area to the right, a turn 
to port must be made on arriving at b by starboarding the helm 
and the turn must be made through two right angles to return 
along the upper side of the line, making a = — -. 

As u = b at the only corner of the w diagram, 

(20) *£ = M (« - b) - * = M {u - b), w - k b = \ M(u - bf 
and it is convenient to take \M = — 1, 

(21) w B — w = (u - b) 2 , 

as before in (25), Lecture I. 

The dotted line bi in the prolongation corresponds to the part 
of the stream line \f/ = along the curved dividing line BI in 
the stream, but this does not form part of the boundary, and 
along it, w B — w is negative, 

(22) u = b + i s '(ic - w s ), 

which is imaginary. 

On the XI diagram, as u diminishes from j to a, 

q = Q,\og—= 0, and 6 diminishes from a to 0. so that the 

line ja is described. 

Passing from a to b, 6 = 0, and ab is described at right 

angles, and extending to infinity, since q = at 6, log — = a> . 

As u passes through b, 6 changes suddenly from to -, so 
that the XI diagram continues in a straight line b a' at a height n 

above ab, and arrives at a' on the line aj a' where log — = 0. 



THRUST AND CENTRE OF PRESSURE 



35 



Beyond a', from a' to j, log — = 0, and 6 diminishes from tt 

to a, so that/ rejoins J, and the circuit is complete. 

Then in accordance with the fundamental theorem (1) of 

conformal representation, 

clQ. . -i _ i _ i 

-j- is composed of the factors (u — b) , (u —a) ', (u — a') ' ; 



(23) 



N 



dn 



du u — b \/(u — a, u — a) 




A' ® 



■«? 




b 




> 

#-7T 


a 


b/- 






tin \ v 




i 



0=0 
Fig. 10. 



a 



In the neighbourhood of u = b, when the chief variation is due 
to u — b, we may replace u — a, a — a' by b—a, b — a', and put 



(24) 



dO = N 1 

da u — b \/(b — a . b — a')' 



and since X2 increases by t> as u diminishes through b f equation 
(14) shows that 



(25) 
(26) 



N 



\/(b — a . b — a) 



dn _ 
du u 



1 I b — a .b — a' 

— b v u — a . u — a'' 



D V 



u — a 



36 THE DYNAMICS OF MECHANICAL FLIGHT 

Integrating 

(27) q = sch- 1 J a - h ; u - a '=2*\r 1 J a '~ h ; 

V a — a . u — o V a — a . 

\/{a — b . u — a') + \/{a' — b . u — a) 

= A l0£J ; 1 7~\ 

& y {a — a . u — b) 

i \/{a — b .u — a') + \/(a' — b .u — a) 
\/ (a — b .u — a') — \/(a' — b . u — a) 

by theorems, of the Integral Calculus, which ought to be familiar 
to the student of this subject. 
At u =j, £2 — ai, 

(28) ch*ai = cos \a = A ~ h 4 ~ *' 

V a — a .j — b 

t, i • • * i • fb — a' . j — a 

V a — a . j — b 

and it is convenient to take j = oo , making 



(29) 


cos ^ a = a / ; sin Jr a = . 

V a —a' ^ 


/& - a 

la — a n 


(30) 


ch - 1 fi — cos *■ a / sh * P - 


- sin \ a 


55 " " * "V u — b 1 *" 


sm 2 u^ 


(31) 


ch Q = ch 2 | O + sh 2 £ O 





COS a + Sin- a 



26 — 6" 

as stated before in (25) I. ; also 

(32) -Q d *=t=Q e 9l =e* 

v ' aw a 

r i lu — a' . . , la — n~\ 2 

= L cos * v ^tf + sm * v^&J 

Over the plane .4.4', a > u >a',6 = 0, cte = dx, 

(33) Qtte= - £ dw 

= " [cos i V^T + Sin * V^f J 2 (6 " " } ** 

= 2 [cos ^ a \/ (u — a) + sin^-a -/'(a — n)]' 2 du, 



THRUST AND CEXTEE OF PEESSUEE 37 

and integrating with respect to u, from a' to ft, 

(34) Q. AA' = f a [2 cos 2 J a (u - a') + 2 sin 2 \ a (a - u) 

+ 2 sin a a/ (a — %. M — ft')] rfzt 
= cos 2 J a (a — ft') 2 + sin 2 J a (a — a') 2 + sin a J 7r (ft — ft') 2 , 

since the last integral represents graphically sin a times the 
area of a circle on the diameter ft— ft' ; and 

(35) AA' = (1 + J » sin a) (ft " a>)2 . 



To express any length such as A'P = x, as a function of w, 
it is convenient to introduce a variable angle <j> (not to be confused 
with the velocity function (fi ), as shown in the diagram (Fig. 11); 

(36) ft — u — (ft — ft') cos 2 $<f>, u — ft' = (ft — ft') sin 2 ^ <£, 

ft^ = (ft — ft') 1 sin <£ d </>. 

(37) ©d# = (ft - ft') 2 sin 2 \ (a + <f>) sin <f> d <£ 

,q ft , ft'a? _ sin 2 ^ (a -j- </>) sin </> ft 7 <£ 

( j IZ 7 " 1 + £ 7T sin a 

_ J sin <£ — \ cos a cos <£ sin <£ -f -| sin a sin 2 <£ , 
1 + J 7T sin a 

,oq\ -P-4' £ (1 — cos <£) — J cos a sin 2 <£ + J sin a (<£ — sin </> cos <£ 
^ ' XT = " 1 -f \ tv sin a 

and is obtained as in (41); Lecture L, by putting <f> = a. 

As the velocity diminishes from Q on the skin of the jet at 
atmospheric pressure to q something less in the interior, the 



38 THE DYNAMICS OF MECHANICAL FLIGHT 

Q 2 — a 2 

dynamic head diminishes by — - . and the static pressure head 

J 2 g ' L 

increases by the same amount ; so that the gauge pressure in the 

interior, the excess over atmospheric pressure, becomes 

1 ' 2 Cg ~ 2 Cg V Q> J 

Along. 4.A', = (Fig. 12) 
„„ j, Q r . / u — a' , ■ , la— u~\v 



q , lu— a • . a — iC\~ 
-±- — 1 COS * a. / =- — sin | a a A. =- I , 



(42) Q dx = - J?dio = £ 2 (« - 6) du, 

q q 

and the thrust dT in the length dx is given by 

<"» »-i%( 1 -$)<-^( 1 -©f'«'- , !f 

5 sin a y (« - ?£ . w - a') ck 



Cg 

_ 



sin a (a - a') 2 sin 2 rt 7 ^, 



2Cg 
and integrating for the thrust T (PA') over P.4', 

(44) TiPA') = -rQ- sin a (a - a') 2 (<f> - sin <£ cos <j>) 

4 C(/ 

(45) T(.L4') = — %- sin a (a - a') 2 77 
v 7 4 Gcj 

(46) ?W) = -5l insula _^ lb £t? of ayerage pre ssure, 
v y A4' Ctf 1 + i 7T sm a 



(47) 



T(P^') _ <f> - sin <£ cos <£ 

IpT) " 7T 



(48) _jT_ = l -.008-2*^ 

^ ^ T(^4') 7T v 



THRUST AND CENTRE OF PRESSURE 
Taking moments round A' to determine L, 
(49) 

(50) TF=\2Z> 



39 



LA'.T(AA') = f^ZT, 






f 77 

= : |i (1 — cos d>) — i cos a sin 2 <A 

1 + jTrsin aj L - v 



jNn 1 — cos 2d> 7 , 
+ ^ sin a((f> — sin <p cos <£)J — - a </> 



= ; (1 — cos d> — i cos a sin 2 <f> + i <f> sin a 

1 + 1 tt sm a J v ^ 2 v i 2 ^ 



— \ sin a sin <£ cos <j> — cos 2 c/> + cos </> cos 2 <£ 
+ | cos a sin 2 <£ cos 2 <jy — \ sin a <f> cos 2 <£ 



+ \ sin a sin <£ cos <f> cos 2 0) 



dj> 

"2^ 




Fig. 12. 

Omitting the terms which vanish by inspection, 

LI' C 77 

(51) -j-jj (1 + J it sin a) = (1—2 cos a sni2 <£ + i </> sin a 



J cos a sin 2 <j> cos 2 <£) 



d 4> 



= (t — J 7r COS a + J Ti" 2 sin a — 1 7r COS a) ~ 2 7T 
= 2 + I w s ^ n a — Te cos a 



(52) 
(53) 



LA' 
AA> 

FL 
AA 7 



COS a 



7r sm a 



COS a 



1 + j 7T sin a 



40 THE DYNAMICS OF MECHANICAL FLIGHT 

This last operation for the determination of L is clue to Lord 
Rayleigh ; although not so very formidable after all, it must 
have appeared so to the inventor working out these integrations 
for the first time with no knowledge of the result. 

The three integrations, of thrust T, length x, and centre of 
pressure L, are typical of what is required in the other extensions 
of this Kirchhoff problem, which follow in Report 19. 

A change of 12 into 7zX2 will give the fluid motion past the 

barrier A A' when it is bent at B to an angle -; and then 

n 

ABA' may replace a cambered wing, touching at A and A'. 

But now the integrations for A'B, BA become intractable, 

and so also for the thrust T, because 

(54) P = [cos J a^/^l' + sin J V^!] 3 , 

(55) Q^ = 2(u-b)C 

= 2 (u — b) n [cos | a V(u — a') + sin | a V(a — u)]n. 

The lift of an aeroplane ^1^4' arises from the gauge pressure 
on the under face, due to the defect in velocity q, and the opening 
out of the stream lines in Fig. 12. 

The pressure in rear of the plane in its wake is taken as 
atmospheric, up to and along the skin of the bounding stream 
A J and A'J r , the air behind being at rest relatively to A A', or 
in a state of vortical turbulence at average atmospheric pressure. 

Any thrust on the plane ^4^4' must be due to an excess of 

pressure over the atmosphere, gauge pressure so called, not 

absolute, on the under attacking face ; this gauge pressure arises 

from a diminution of velocity of q below Q, and diminution of 

Q 2 a 2 
dynamic head, from ~^— to ^- , and an equivalent rise of static 

pressure head, due to the broadening of the interval between the 
stream lines close to ^4^4', especially in the neighbourhood of B, 
the branch point. 



THEUST AND CENTRE OF PRESSURE 



41 



This shows in a general way why L, the C.P. (centre of pressure) 
is away from F, the C.F. (centre of figure) and more towards B, so 
that the tendency of the fluid reaction is to turn the plane more 
broadside to the stream. 

A popular figure of the stream lines past a cambered wing, as 
here in Fig. 13, showing no such broadening, would imply at 
once to our eye an absence of all thrust and lift ; the figure 
should be more like Fig. 14. 




Fig. 13. 




Fig 



In the electrical law of flow where there is no wake, as 
here in Colonel Hippisley's diagram (Fig. 15), the broaden- 
ing of the stream is shown near a branch point ; but as 
the stream lines close in behind symmetrically, there is no 
resultant thrust on the cylinder by the stream tending to 
move it, but a couple only tending to turn the cylinder broad- 
side to the current, arising from excess of pressure near the two 



opposite branch points. 



4-2 THE DYNAMICS OF MECHANICAL FLIGHT 

The magnitude of the couple on an elliptic cylinder in found to 



be 



Q* 



sin 2a times the weight of the column of liquid with a 
"9 



circular base on a diameter joining the foci. 




Fig. 15. 

The couple is shown experimentally by letting a card drop in 
the air from the horizontal or vertical position, or by projecting 
it with a spin, observing how T the card turns gradually into a 
vertical plane. 

So, too, the stability is assured of the forward course of the 
flying machine by making the spread of the wings three or four 
times the depth from front to back. It is supposed, too, that the 
stability can be improved by giving the wings a dihedral angle, 
as at the top of the stroke of the wings of a bird. 

The existence of a couple on a flat or elongated body moving 
in a medium, air or water, tending to set it broadside to the 
motion is discussed in Thomson and Tait's " Natural Philosophy," 
and in the Hydrodynamical Treatises of Basset and Lamb, in a 
treatment rather abstruse. 

But an explanation can be given in the manner called " ele- 
mentary," depending on the principle of momentum. 

Take a body like an elliptic cylinder, as in Colonel Hippisley's 
diagram (Fig. 16), and supposing it moving with velocity Q, and 
velocity components U and V, being held so as to prevent 
rotation about the axis of the cylinder. 



THRUST AND CENTRE OF PRESSURE 



43 



If there is no surrounding medium, the components of 



u 



V 



momentum, ] - and W sec-lb of the body weighing 

.9 9 

W lb will remain unaltered while the centre moves from to 
0', so that the vector OH of momentum will move to 0'H r 
in the same straight line 00', and there is no change of 
momentum and no force or constraint is required. 



y 




c 2 wf 


7 


F' 






^ 








^^_^t) 




^x^ 


v. 




-S^^ 










C 




u^y 


X 









momentum-components in the medium, 



Fig. 16. 

But if the body is surrounded by a medium, of which it dis- 
places W lb, the velocity components U and V will give rise to 

ciW — , c%W — , where 
9 9 

Ci and e 2 are different constant numbers depending on the shape 
of the body ; and as the broadside motion will give the greater 
momentum, c 2 >ei. 

The body must now be held to prevent rotation ; because at 
0' the momentum of the medium has changed from OF to O f F f , 
not in the same straight line 00' ; and the change of momentum 
is the impulse couple iVi = OF . OD, acting on the medium, 
against the clock, OD being the perpendicular on ()' F' . 

The medium reacts on the body with an equal and opposite 
impulse couple Xi, tending to set it broadside ; and if t seconds 
is the time from () to ()\ the impulse couple given by its 
components against the clock is 

(56) N t = c 2 W -. Ut - c,TF -. Vt 



9 



= {c. 



1 * ? ^ 



9 
ft-lb-sec. 



44 THE DYNAMICS OF MECHANICAL FLIGHT 
the accumulated effect in a time t of an incessant couple 

(57) N = (c 2 - Cl ) W y, ft-lb, 

and this is the expression found in Hydrodynamics. 

For an elliptic cylinder, as in the diagram (Fig. 17), it is 
found by theory that 

(58) Cl = -■ c-^i 

and for a length I ft in a medium of density w lb/ft 3 , 
W - TTwabl, 

(59) 2r=(-_£).. a »,!n: 

uv uv 

= irw (a? - V) I — - = W" , ft-lb, 

v ; 9 9 

where W" is the weight of medium displaced by a cylinder of 
cross section the .circle on the diameter SS' joining the foci 
S and ,S". 

This reduces for a card of breadth 2a, with b = 0, to 

UV Q 2 
(60) N = iTiva-l — = 77- Acitt sin a cos a, 

9 u 9 

assuming the electrical law of flow round the edge A and A'. 
But this would be qualified by the factor 

3 
SI 



(1 + \ 77 sin af 



in Kirchoff's discontinuous motion, where the couple, in accor- 
dance with (53), would be 

(61) 77- Acltt sin a COS a 



Cg (1 + J 7T sin a) 2 . 

The calculation is not simple of ci and c 2 for other figures, 
such as a solid of revolution ; but it is required in the discussion 
of the stability of an air ship or submarine boat, in its effect 
on steering and loss of metacentric height. 



THRUST AND CENTRE OF PRESSURE 45 

In artillery the theory is useful for determining on gyroscopic 
principles the spin requisite for stability of an elongated shot 
fired from a rifled gun. 



Fig. 17 



When the external shape is assimilated to a prolate spheroid, 
generated by the revolution of an ellipse of axes 2a, 2b (Fig. 17), 
and when the length in diameters 2a/2b is denoted by x, we 
find by hydrodynamical theory, on the electrical law of flow, 



(62) 



A B 



C, = n F' Co = 



1 - A ^ 2 ~ 1 - B 
where A and B are determined from the equations, 

(63) A + 2B = 1, x*A + 25 = x sh ~V C* 2 ~ x ) 

v 7 ^ 2 - 1) 

= v(J g -l) l0ge[ * + v^ 2 -l)] 
changing for an oblate spheroid, # < 1, to 

.r sin-" 1 V(l — ;r 2 ) 

V(l - x 2 j 

For a sphere, # = 1, A = B = J, d = c 2 = i, and the 
effective inertia of a sphere is increased by half the weight of 
the medium displaced. 

But for broadside motion of a cylinder, x = oc, A = 0, 
B = J, c 2 = 1 ; so that the inertia is increased by the weight 
of the medium displaced. 

Thus a spherical balloon or cylindrical air ship of weight IT lb, 



46 THE DYNAMICS OF MECHANICAL FLIGHT 
displacing W lb of air, will start to rise with acceleration 
W -W ^ W - W 



(64) 



vv — w \v — vv 

i~W + W g ' or W + W g ' 



A spherical bubble of air in water, w T here TI r is insensible 
compared with W, will start up with acceleration 2g. 

A flexible sheet in tension t, separating the current of density 
w from a dead wake of different density w\ as with water past 
air, and flapping like a flag or shivering as a sail, would swing 
in waves of length A advancing with velocity U, such that in 
accordance with wave motion theory 

(65) ?^1 = w(U - 0)2 + «,' U-2 

A 
U = — n / r ^ * WW ' TT2 "I 

to + w' + * [_(iv + io')\ ~~ (w + w') 2 ^ J 
so that the waves break up into vortex motion of 

(66) -^— < : J L 2 . 

v ' a w + w 

The practical question arises as to the advantage of a short 
length of fringe or flap at the edge A of the aeroplane, so as to 
retard the formation of a vortex, such as seen at the lee of the 
chimney top. Some such arrangement can be seen in the new 
flying machine. 

In the absence of these wings the equipotential lines of the 
electric field would bend round the barrier symmetrically (as in 
Fig. 18), and IB would be the branch of a hyperbola with foci at 
A and A', continued on the other branch B' V at zero potential. 

This electric field would represent the analogous streaming 
motion, realised by coloured filaments in Hele Shaw's experi- 
ments, in a viscous liquid, where the motion is slow and no 
perceptible eddy is formed at A and A'. 

The liquid is then said to stream on the electrical law of flow ; 
and the more general case where A A' opens out into a confocal 
ellipse is shown in Colonel Hippisley's diagram here (Fig. 15), 



THKUST AND CENTRE OF PRESSURE 



47 



which can be interpreted electrically as representing the dis- 
turbance in a uniform field by the presence of an elliptic cylinder 
to earth. 

A magnetic interpretation can also be given to his diagram as 
a generalisation of Maxwell's Figure XV., of a magnetised 
cylinder in a uniform magnetic field. 




Fig. 18. 



It is gratifying and of great assistance, too, when the mathe- 
matical analysis of a mechanical question can be made to 
serve in another interpretation ; and so the student of electricity 
should be interested in following up the electrical analogue of this 
hydrodynamical problem. 

We must suppose the uniform horizontal current to represent 
a uniform vertical electrical field, and this field to be disturbed 
by the uninsulated plane strip A A' ; while the skin stream lines 
A J, A' J' must be replaced by wings of flexible gold or tin foil, 
as in the electrometer, but stretching away to infinity. 

The wake of the plane AA', which to the pilot seems stationary 
behind him, is being dragged along through the air by the flying 
machine. 

Any machine following which flies into this wake or backwash 
will seem as if it is entering still air ; the lift is lost and the 
machine will drop. This may happen if the pilot is following 
at a different level, above or below, so that in passing another 
machine it is prudent for the pilot to take a course to one side. 



LECTURE III 

HELMHOLTZ-KIRCHHOF THEORY OF A DISCONTINUOUS STREAM LINE 

The normal conditions of the aeroplane of a machine flying 
high irp in the air are represented on the Kirchhoff theory by the 
diagram 4 and 10, representing the state of flow relative to the pilot ; 
the analytical conditions have been discussed in Lecture I. and II., 
and will be found in § 6 of Report 19. 

But there is an advantage in extending the theory to a more 
general case, and beginning as in Report 19, § 2, with a plane 
barrier oblique to a stream of finite breadth (Eig. 19), as the 
extension is not essentially more complicated, and it throws 
light on the simple case of an infinite stream. 

No change is required in the 12 diagram (Fig. 19), except that 
i, j, j , on ad are now distinct ; but now in the w diagram the 
outside stream lines come from infinity into view, and the 
diagram has the advantage of being closed, as in Euclid I. 27, so 
that the difficulties at infinity are kept under observation. 

In the w diagram there are three stream lines in the boundary, 
which we denote by \j/ = mi, m 2 , m^; and \j/ = /»i is the outer 
stream line from I to J ; ^ = m. 2 the inner skin from J to A, from 
A along A A' to the branch point B and then to A' , and again 
along A' J' ; \}/ = m 3 is the other outer stream line from J' back 
to I. 



THEORY OF A DISCONTINUOUS STREAM LINE 49 



There is also the curved branch IB of the \ff = m 2 stream 
line along -which the current divides ; but as IB lies in the 
interior of the fluid, and the velocity q and its direction 6 both 
vary along IB, the corresponding part ib in the w and 12 diagram 
must be excluded from the boundary ; ib is straight in the w 
diagram, but curved in the £1 diagram, from b at midway height 
\- of a and a', to i at height a. 




Fig. 19. 



The branch point B where the velocity is zero requires 



(1) 



— at u = b, so that, by II. (1), 

dw u — b 

du ~ w — j.u — j'.u — i 



m 2 — w 3 _ m s — m x 



u -J 



h — m 2 t 1 1 \ 

tv \u —j u — i) 



+ 



u — I 
m — in. 



u 



H 



when resolved into partial fractions, ^Yith the factor appropriate 
for making w change suddenly by mi — m- 2 , m- 2 — m s , m 3 — wii, 
as u passes through j, j', i, in accordance with IT. (15). 

M.F. E 



50 THE DYNAMICS OF MECHANICAL FLIGHT 

Integrating, 
(2) iv + a constant = — - 2 log J -. H 2 - ? log 



% — 1 
and with origin at B, 

(3) w - to* = -i - 2 log i . . 

7T U — I O — J 

+ — log i- • t . > 

it m — 1 — j 

and taking i = 00 simplifies this to 

(4) W — 10 B = — i ? log i + -J 5 log - -L-. 

ir b - j ir b - j 

- The z diagram is now as in Fig. 19 , and if c lt c 2 denotes the 
breadth of the stream at J and J' ', where the velocity has become 
uniform and equal to the skin velocity Q, 

(5) Qci = m l — m. 2 , Qc 2 = m 2 — m 3 ; 

and putting c\ + c 2 = c, the breadth of the impinging stream 
at I, 

(6) Qc = m A — m 3 ; 

and the curved part BI of the dividing stream line \j/ = m 2 
separates the stream at I into two parts of breadth c\ and c 2 . 
Then we can write 



IV — u 



1 u — j b — i , -, u — 1' b 

(7) 7T -— * = c 2 log < . : + c 2 log 4 . 7 - 

v (*/ u — 1 b — 3 u — 1 b 

and at 5, where 

(8) u = b, g . 0, 

v> - j ft - v \v- j u - *; 

and with the sequence 

i (00 ) > 7 > « > & > «' > ;' ' > « (— °° ), 



THEORY OP A DISCONTINUOUS STREAM LINE 51 
1 ,+ , 1 



/w e 1 b-j' i-b _ i - j' . j - b 

(9) 7, = 1 _ 1 -i-j ■ b-f 

j — b i — b 

With the value of X2 as before in II. (27), we must write in the 
region i > u > a, 

(10) n = log 2 + oi 

o- ~ l /a — b . u — a' . . _j /6 - a' . u — a 

= % cos / -. j- = M sin i A / ; r 

\J a — a . u — b \ a — a . u — o 

so that log — = 0, q = Q, as required over the skin of IJ and 
J A ; and 



,., ., v 9 1/1 a — b . u — a - 9 -i a b — a' . u — a . 

(11) cos 2 i = T , sm 2 \ 6 = ■, T ; 

v ' * a — a' . u — b a — a . u — o 

and \jf being constant, 

(12) *£ = ** = ^* _o£» = constant-^; 

aw aw as aw ^ aw 

and thence the intrinsic equation is derived of I J and J A, as 
given in § 2, Report 19. 

The linear scale of the diagram is calculated as explained in 
§ 3 by an integration, expressed in terms of the arbitrary 
constants a, a', b,j,j\ i, in a transcendental form, from which 
the length AA\ AB, BA' is calculated, and the angle a, f3, 
fi f , at which the stream from I is received by the plane and 
leaves it again at J, J'. 

The inverse process to determine the constants a, a', b,j,f, i 
from the dimensions of the diagram would be intractable 
analytically, and would require the trial and error process. 

The thrust T is obtained immediately in this case, and with- 
out integration, by the principle of momentum ; resolving 
perpendicular to the plane A A/ 

e 2 



52 THE DYNAMICS OF MECHANICAL FLIGHT 



(13) T = (m, - m 3 ) §- sin a - {m x - m 2 ) §- 



Q 

— (m. 2 — m 3 ) jr- sid/S 



og 



sin 



91 

Cg 



(c sin a — Cj sin/5 — c 2 sin/3'), 



Eesolving parallel to the plane 
(14) = (m, - m 3 ) q- cos a - (m 2 - mj ^- cos /3 - (m. 2 - m 3 ) 



Q_ 

Cg 



cos 



(15) 



— c cos a — Cj cos /3 — c 2 cos /3', 



which is found to verify. 

But the determination of L the C.P. is intractable for a 
stream of finite breadth. 




Fig. 20. 



The extreme case of an infinite barrier A A' receiving the 
impact of a jet of finite width is obtained, as shown in Fig. 20, 
by making j = a,f —a'. 

Duplication in A A' with the motion reversed will give the 
direct impact of two unequal jets with equal velocity, as in 
Fig. 21. 

A simple experimental illustration can be given of these 
problems with a blade held in a jet of water from a tap, or with 
impinging gas jets. 



THEOBY OF A DISCONTINUOUS STREAM LINE 53 

To return to an aeroplane high up in the air, in a relative 
current of infinite breadth, we must take i, j, f equal, and the 

integration of -7— leads to an algebraical relation for w, not 
du 

logarithmic ; 

(16) — - = M-, T *,w 

v ' du {u — if 

and writing \ M = (i — 6) 3 , 

(17) w - w B = (u - by 



1 M (u - 



1 M (u - b\ 2 



u — 1 



and then, as in (21) II., by taking i = 00 , 

(18) zv - w B = (u - b)\ 




Fig. 21. 

But it is quite as simple, and more instructive, to consider a 
general case, especially as we are then at liberty to alter the 
sequence of the arbitrary constants, and so make the same 
analysis serve for different hydrodynamical problems, hitherto 
considered as distinct. 



Previous writers on this subject strive to give special values 
of these arbitrary constants, such as Q = 1, and a = + 1 at A, 
a' = — 1 at A 1 ; but, as we shall see, at a loss of flexibility in 
the analysis. 



54 THE DYNAMICS OF MECHANICAL FLIGHT 

The only special value we have adopted so far, is i = oo 
occasionally, but better only after the general case has been 
examined, as the passage to i = oo is sometimes treacherous. 




Fig. 22. 



B( A 




Fig. 23. 




Fig. 24. 

To illustrate the flexibility of the analysis, we examine the 
interpretation of (4), (10) due to a rearrangement of the 
constants, in the order 

(19) *'(oo)> a >j > a! > b > j' > i (- oo). 

There is no alteration in the w and 12 diagram and relation, 
but the z diagram changes as shown in Figs. 22, 23, 24, where 
the barrier A A' is replaced by a slit or leak made in a plane 
boundary. 



THEOEY OF A DISCONTINUOUS STREAM LINE 55 

If the stream past the slit is very broad, we make j' = i, and 
m 3 — x , as in § 14, Report 19. 

These extensions give an insight into the subject from the 
point of view of generality, without additional difficulty, and 
make the initial problem of Kirchhoff appear much easier. 

In continuation of these extensions, bring the blade in 
the experiment close to the tap to imitate the motion 
when the efflux from a channel is received on the plate 
(Fig. 25). 




k ® 



/ 


!_ 


a 


>' u 


\ 




a' 


/f 




A' 




Fig. 25. 



Then if K, K' are the ends of the channel in the z diagram, 
the 12 diagram is modified by an inset kik' ; so that with four 
right angles at u = a, a' , k, k', and a zero angle at b, and a turn 
to port through two right angles at i, 

nn\ dn at " ~ J 1 • 

die u — o v ( u — a • u — a . u — k . u — k) 



and with 

(21) 



N 



(« - ^) — = 1, at u = b, 

du 

x(b — a . b — a . b — k . b — k') 



56 THE DYNAMICS OF MECHANICAL FLIGHT 



so that, writing 


(22) 


u — a . u — a . u — k . u — k' = U, 




b - a.b - a' .b - k . b - k' = B, 


(23) £<° . 
du 


u - i */B \ s/B 1 \/B 


' u - b .b - iyU ~ b — i */U + u - b JU 



introducing the elliptic integral of the first and third kind 
(I. E.I. and III. E.I.). 

But there is no alteration in the iv diagram and relation of 
Figs. 10 and 19 ; and as we proceed to problems of increasing 
generality, the diagrams may change one at a time, of ic and n. 






K ^ A 



? 



Fig. 26. 



The barrier A A' may be placed inside the entrance KK' of 
the channel, as in Fig. 26, representing a rudder boxed in. 

By making j = k, f = k r , the barrier becomes of unlimited 
length, with a barrier A A' across it aslant. 



J' 



J- 




FlG. 27. 

Further, by taking j' = k' = i, the upper barrier IK' is at 
infinite height, as in Fig. 27, and the analysis will serve for an 
aeroplane flying horizontally near the ground, as in making a 
start in flight. 

The analysis is seen to be too complicated for practical 
application, but § 11 and § 59 of Beport 19 show how a solution 
can be obtained of quasi-algebraical character, when the barrier 
or rudder AA' is set at an angle \ ir/n, an aliquot part n of a 
right angle, the III. E.I. now becoming pseudo-elliptic ; and for 



THEOEY OF A DISCONTINUOUS STREAM LINE 57 

the identification of results it is convenient to take k' = x , 

'b — a . b — a' . b — k 



(24) 






u — a . u — a , a — k 



A biplane machine, with two decks, AA', KK', would have 
the z diagram, with w and 12 as in Fig. 28, and then, with 
\fr = m, m' over AA' , KK' 




/J 



® 



b<<b' @ j- 

\\ . k 



k' 



Fig. 21 



IC%K . dw ^ r u — b . u — b m — m fit -\- i — b — b 1 

(25) = m , = . J — : i : 

die u— j 7T \ j — b . ; — b ~ n — ] 



(26) m — m' 

_ 1 (u + k- 2b)(u + k_- 2b') -(a + k- 2b)(a ±Jc_- 2b 1 ) 
U - b )U - b') 



+ log i 

<>-3 



(27) 



N 



^O _ _ 

du u — b . u — b' ' \/{u — a . u — a' . u — k . u — k') 
1 /B 1 /& 

~ u - b V U ~ u - b' \ U 



introducing the III. E.I. again. 

Experiment with a fork, two-pronged, held under a tap. 



5S THE DYNAMICS OF MECHANICAL FLIGHT 

In the short time at disposal in these Lectures it is not 
possible to follow the elliptic integral interest much further ; 
but these aeronautical applications are calculated to give a rapid 
insight for those who wish to study the subject. 

Thus in § 37, Report 19, an application is made to the vortex 
in a polygon, an eddy whirlwind, such as Chavez would have 
to negotiate in a corner of the precipices, when crossing the 
Simplon, 1909. 

The analysis is the same as that required for the electric field 
of a prismatic condenser, of which the cross section is bounded 
by concentric similar polygons, with a dielectric interspace ; and 
it is the mean equipotential surface which corresponds to a free 
surface of the vortex in a polygon (Fig. 29). 

A single vortex travelling parallel to the ground, 
or a pair of such vortices duplicated by reflexion 
in a plane, will have a curious vortex sheet 
surface, such that any point on the surface will 
trace out an Elastica, the curve made by bending 
a straight spring ; much as a point on the rim 
of a wheel will trace out a cycloid over the road, while a particle 
of the tire describes a circle relatively to the carriage. 

The trail of straw and paper of a windy day, left on the pave- 
ment near a wall, will imitate the general appearance of the 
curve for a vertical vortex, such as the pilot may be required to 
negotiate high up in the air. 

But it is the motion of a horizontal vortex near the ground 
which will increase the difficulty of alighting. 




A 

Fig. 30. 



The hydrodynamical application of conformal representation 
originated with Helmholtz in 1868, when he applied it to the 



THEOEY OF A DISCONTINUOUS STEEAM LINE 59 



efflux of a stream between two parallel walls, extending into 
the fluid, as in Fig. 30, of a Borda mouthpiece. 

But as the general solution is analytically of equal simplicity 
for a stream issuing from a channel between two parallel walls 
FA', IB, blocked partially by a lip BA, as in Fig. 31, we 
resume the consideration of conformal representation with this 
application, and treat the original 1868 problem of Helmholtz as 
a particular case. 




O 



\W) 



a 



a' 



Fig. 31. 



The fluid is bounded by two stream lines, \f/ = and \f/ = m, 
so that the w diagram consists of two parallel straight lines, 
joined up at i and j, as in Euclid L, 27. 

The £2 diagram is open rectangle of f i em i -infinite length, with 
right angles at u = a, a', and stretching to infinity at u = b. 



Then 




(28) 


dw M m 1 m 1 


du ~ u — i . u — j ~" 7T 4 — j 7T " u — i 


(29) 


W " Wb - \oc " 2 - - ; h " * 


" m - 10g : •■ ' b - j> 


reducing, 


when we take i = oo, to ' D 


(30) 


, W - Wli loc U " j • 


m ° b — i 



and a further simplification can be made by taking ; = 0, 



(31) 



60 THE DYNAMICS OF MECHANICAL FLIGHT 

with the sequence 

(32) i (go) > b > a > > a' > •' (- oo). 

For the 12 diagram 

^ ' du u — b ' v (u — a . u — a')' 

with 

IT 

(34) (« - 6) gj = 2 -» = i-, ,vhen « = 6, 

if the exterior angle at B is J ^r/n, one wth of a right angle ; 

rqe\ ^ 1 1 /b — a . b — a 

du In' u — b' \' u — a . u — a" 

and thence, by the previous integration in II. (27), 

(36) £» = e' ;n = a 7 (b - a.u - a') + y x (6 - a . u - a) 
v ; fa N (a - a' .u - b) 



C37) ?n — (9\ n — /^ — rt ' • ft ~~ U 4- /^ — ft ' ft ' '" u 



Along Z'^4', — oo < u < «', = 0, 

? = ('■ , 

\2 / V a — a' . b — u \ a — a .b — u 
Along A' J, a' < u <.j,q = Q, 

(38) p = *« = . / 6 ~ ^ a -^ + i A /*-y ~ a ' 

y J s V a- a - u V a - a . u - b 

and so also along J A,j - a <i a, q = Q ; 
and measuring the arc s from A' , 



(39) Qs = 4 - ^, = to - Z0 A , = ^ log £ 



and if c is the breadth of the jet at J, 

(40) Qc = m, u — a'e " c, 



THEOKY OF A DISCONTINUOUS STEEAM LINE 61 



b — a . u 



which combined with 

(41) cos 2 nO = j — ; , sin-Ht 

a — a . b — n a — a ' . b 

will give the intrinsic equation of A 'J or J A. 
If 6 = (3 at J, where u =j = 0, 

(42) cos 2 «/3 



sin-nfi 



b — a' . a 


1 


~ b 


a — a' . b 


1 


a'' 

a 


b — a . — a' 


1 
1 


a 
~ b , 


a — a' . b 


a 



and if q z denotes the velocity at I, where u = oo, and a 7 the 
distance between the two planes IA, I' A', 




Fig. 32. 

The figure can be duplicated about IB, and so give the 
motion when the channel is blocked by a wedge-shaped pier 
(Fig. 32). 

When A' is carried along on 1'A' up to J at an infinite 
distance, making a' =j = 0, the wall I' A' is of infinite length, 
and the motion may be duplicated again about I'J, and so 
represent the flow of water through the piers of a bridge, wedge- 
shaped as at Westminster. 



62 THE DYNAMICS OF MECHANICAL FLIGHT 

Duplicating the original figure once about I'J' will give the 
efflux from a channel with a conveying mouthpiece, as in Fig. 33. 



Fig. 33. 



Make b = cc, and the flow is obtained through the gap between 
two walls converging at an angle n/n ; and now along the skin 
of the jet A J, of breadth Qc at J, 



(44) 



^ s m -, a 

10 a — xo = Qs = m -^ = — log -, 



(45) 



sin 2 w0 = - = e~ n o 



the intrinsic equation of the jet A P. 

In Helmholtz's original problem, where the liquid is drawn 
off between two parallel walls like a Borda mouthpiece (Fig. 30), 



(46) 



sm- 



ds 
cl6 



cot 



and if d is the outside distance between the walls, 

(47) id - c = I - sin 6 ds = - 2 cos 2 \Q d0 = c, d = 4c, 

Jo ^ J 

so that the coefficient of contraction is J. 

In Helmholtz's next problem, where a slit of breadth d is made 
in a wall, through which the liquid escapes (Fig. 34), 



(48) 
(49) 



1 * 

n = 1, sin 6 = e~ 27r * 



PM 



JVM 



7 2c 
as = — e 



dy 
" ds ' 

2c . 
— sin 

IT 



PT 



1c 



THEORY OF A DISCONTINUOUS STREAM LINE 63 



so that 


the 


curve 


AP 


is 


the tractrix ; 


and 














(50) 






¥ - 

2c _ 


c 


= AO 


_ 2c 
0-611. 




the coefficient of contraction. 

Report 19 may be consulted for appli- 
cation of the same analysis, where the s 
figure changes with a rearrangement of 
the order of the arbitrary constants. 



y-' 



\ 



Fig. 34. 






g 




a 






A 


c 






?I6. 


3.:. 




a 



As a general exercise on conformal representation, consider 
the application to the z diagram in Fig. 35, representing uni- 
plane injector flow, with the associated w and 12 diagrams. 
where the z figure may be supposed duplicated in the median 
line IJ, and here 

j(oo) >&> a > c > a > V > > k >i(-oo). 
Next examine the change in the z diagram due to a rearrange- 
ment of the sequence of the constants 

a, a ' , b, b', k. 



64 THE DYNAMICS OF MECHANICAL FLIGHT 

Digression on the Integral Calculus. 

Leaving the elliptic integral interest as leading too far, and 
passing over the analytical expression of iv, as given merely 

by the algebraical and logarithmic function, since -=- can be 

resolved into a quotient and partial functions integrable imme- 
diately, we concentrate our attention on the determination of 
O from the typical form 



(51) 
making 

(52) 



Returning through Paris in May, 1910, I took the opportunity 
of attending a lecture at the Sorbonne by M. Marchis, the new 
professor there of Aeronautics ; and I found he devoted a whole 
lecture to the consideration of a single integral, which with 
Greek letters we can write 



dn _ 

du 


i n, 

u — b V u 


— a . 

— a . 


b 
u 


- a 1 

- a< 


(«. 


7\ dQ, 


= 1, 


when 


u 


= b. 



(53) = J_ 



pd$ 



+ y cos 



employing a normalising factor ft, which is either 

^(a 2 - f) or yV - " 2 > 

But as M. Marchis was allowed, not six ; but some twenty 
or thirty lectures, he could go thoroughly into this detail, 
essential in the theoretical study of the aeroplane. 

Marchis' s integral is identified with the integration required 
in (27) II. for 12 by putting 

(54) a — u = (a — a') cos 2 J0, u — a' = (a — a') sin 2 ^0, 
\/{a — u . u — a') = J (a — a') sin 0, 
du = J (a — a') sin 6d0, 
du 



^/(a — u . u —a') 



de, 



THEOEY OF A DISCONTINUOUS STREAM LINE 65 

a + &' — 2z^ = (ft — a) cos 6, 

b — U = b — 2 ( ft + a ') + 2 ( a — a ') cos = a + y COS #, 

a = 6 — | (ft + «'), y = i (a — ft'), 

a + 7 — &' — a' , a — y = b — a, 

/3 = v'(^ — a • 6 — «')> 

* = f v(5 -a.b- a') Pit, = nu 

)(b — u) \/(a — ft . u — ft) 
Or in the hyperbolic form, 



(55) ft = V{ f - *), * = \- {F ^ 



6.6- «') fi _ 



/ (ft — ft . U — ft ) 

In a complete treatment of integration the hyperbolic function 
ranks equally with the circular function, direct and inverse ; 
especially in this subject of conformal representation and its 
hydrodynamical application, where the change from circular 
to hyperbolic form, and back again, is taking place continually. 

Thus, if 

(56) u — ft. u — ft' is positive 

(57) oo > it > a, or a' > u > — oo , 

(58) u — a = (ft — ft') sh' 2 J v, or — (ft — ft') ch 2 \ v, 
u — ft' = (ft — ft') ch 2 | v, or — (ft — «') sh 2 | v t 

*J{u — ft . ft — ft') = J (ft — ft') shu, 

c/'^ = \ (ft — ft') sh v dv, 

clu 7 
; —r = »#> 

y^-ft — ft . U — ft ) 

2m — ft — ft' = + (ft — «') ch v, 

b — u = b — \ (ft + a ) + 2 ( rt — rt ) cn v > 

= a + y ch V 

a = b - \ (ft + «'), y = + \ (a - a), 

a — y — b — ft', or b — a, 

a + y = b — a, or b — ft', 

(59) p 2 = a 2 - 7 2 = b - ft . b - ft', 
v/(fo — a.b — ft') f /? c/v 



(60) f v(h ~ a ■ ° ~ "' - = f- 

J (0 — «) v" ( u — a . u — a) J a 



+ y Ch V 
M.F. F 



66 



THE DYNAMICS OF MECHANICAL FLIGHT 



The contrast is brought out in the geometrical interpretation 
by means of the ellipse and hyperbola in Figs. 36, 37. 

Changing back from the Greek to italic letters, to agree with 
the ordinary notation, we take 



N 




Lo 


A 


K 


L 


/ 






C 


t 


' 1 


V A 


X 



Fig. 36. 




GA = a, GF = c, CX = °L 

c 
tf = a 2 - c 2 



FP 


= r 


V 




a -\- c cos 6 


<£ 


-j 


r bcie 


a + c cos 


4> = 


COS" 


_ : c + a cos 


a -\- c cos 6 




= sin - 


_! b sin 
a -f cos 6 


"With 


c = 


a COS a 


tan 2 1 <£ 


_ 1 — COS cf) 

1 + cos </> 




a — 


5 1 — COS # 



ft + c 1 -f cos 
tan 2 i a tan 2 J 



C.4 = a, Oi^ 7 = c, CX = 






= c- - a? 

g 

ft + c cos 

r bde 

a + c cos (9 



<£ = ch-i c ± a CQS 
a -\- c cos 



With 



sh 



th 2 I <£ 



_! & sin 



ft + c cos 

ft sec a 
_ch<£ - 1 
~ch</> + 1 

ft 1 — cos 



c + a " 1 -j- cos 
= tan 2 ^ a tan 2 ^ 



= ^P=^.Pir=^(CX-Cilf) r=^P = -£p^ = £_(CJf-CX) 

= -(— - ftcos</)j=ft-ccos<^> =- (a ch<f> - ~) — c ch - a 



a + c cos 



ft + c cos 



THEOEY OF A DISCONTINUOUS STREAM LINE 67 

CM = x = a sin <£ CM = x = a ch <f> 

MP = y - b cos <£ ifP = y = b sh <£. 

Circular sector ACp = J a 2 </> Hyperbolic sector ^4CP = \ ab 4> 

Elliptic sector ^4CP = | a& <£ Hyperbolic sector AFP = \ ac 

Elliptic sector AFP = \ ab<f> sh <f> — \ ab <£ 

— \a sin <£ 

Mean anomaly ?z£ = <£ - e sin <f>, Mean anomaly 7^ = e sh <£ - </> 
in the elliptic planetary orbit, in the hyperbolic orbit ; 
S the true anomaly, <j> the 
excentric anomaly. 

Also, for the integral (60), 
dO a + c cos b 



dcf> b c ch <f> — a 

a _ f 6<Z<£ _ 2tan -ithi<£ 



J c ch <£ — a tan ^ a 

(a = c cos a, b = c sin ft) 

-K-ffl ch <f> . _1 6 sh d> 

= cos — ~ = sin 



c ch <£ — a c ch <f> — a 

With # as the variable we may take as the typical irrational 
integrals which occur 

<61) ■ Jf^* 



- v) y 

y = -v/(^ 2 + 2foc + c), 
in which the integrations are said to be taken round the conic 
(62) y* = ax 2 + 2bx '+ c. 



The most general irrational integral of this nature 

dx 

y 

where P and Q are rational fractions, the first integral being 
required for w and the second for 12 in the previous applications. 



(63) I </> (x, y) dx being reducible to \ Pdx + j Q 



There are three regions to consider, according to the shape of 
the curve of (62). 

f 2 



68 THE DYNAMICS OF MECHANICAL FLIGHT 



c 

ft 



550 

CD 



02 

o 

ft 
e 





c3 




CD 




Jh 




, , 








c3 




f-i 


CD 


O 


> 


«H 






to 


CD 


CD 


*+= 


fl 




O 


CD 


e 


S 


1 


CD 


!N 


r^ 






„ 


2 


CD 




.£ 


3 




o 


°-J-= 

• i— i 


£ 


o 








1—4 


CM 


8 


O 






o 




tt 


+ 


-a 


+ 


~^ 


-~ss 


ft. 


C<l 


1 




+ 


Q 


+ 


1 




ft 



r> > 



+*? 







«; 




V) 


s 


+ 




+ 


; « 




Ci 




« 1 


r^<s 


3 


ft 


C5 


«i C<J 


1 


c5 


•s. 


+ + 


<N 


+ 


^-. 


a in 






^ 


^^ 




ft 




■N « 




Q 



> 



> > 



5* I >, I 



H 



OS 
CO 



to 



3 








CD 






.£ 












*-4-=> 






c5 




- — -, 


bD 


!>^ 


o 


CD 


f-i 


^ 


H 


c3 


tc 


^ 


*5c 


s 


4-= 
o 


c3 




^Q 


— 


c3 




• i—i 








C 


o 


o 


,0 


^ 


'3 


Cp 


1 


o 




1 


u 


CD 


cq 


CD 

4-3 


C3 


^5 


r^ 


CSD 


fl 


fl 


£ 


c3 


c3 


• i — s 












O 


Q 


H 


O 




S4H 




rSH 


o 








HH 


• i— 1 


02 


1— 1 


£ 


CD 
> 



I J 





















rC 


| 


rC 


(M 


<N 


1 


N 


1 














1 




1 



> > 



s. I 





M 


— \ 

1 


I 
ft 


> 


> 


1! 


^'r 


— 


1 



THEOEY OF A DISCONTINUOUS STREAM LINE 69 

where it will be noticed that the second integral is circular or 
hyperbolic according as if is negative or positive for x = p, and 
the multiplier M is either 

v / (- af- - 2 bp - c) or s /(ap- -f 2 bp + c). 

In I. and II. y 2 can be resolved into factors, x — a, x — a' 
suppose ; and replacing a by — 1 or + 1, the result can be 
simplified and the integral corrected for one limit. 

I. a > x > a . 

f a dx 



2 sin- 1 



/a — X ! X 

. / ? = 2 COS -1 . / — 

V a — a v a 



J n -v/( a — x • x — a ) v a — a' V a — a 

f ffo t /a — £ • , lx — a 

\^— f r- = 2 cos" 1 / = 2 sin- 1 /- — , 

J a \/(a — X . X — a) V a — a V a — a 



rfx 



i' \/(a — £ . X — a ) 

II. ao > x > a. 



P ^_ __ = 2B h-i /^^ = 2ch-i /^A, 

J a -/(# — a • x — a ) \ a — a 'V a — a 

a > .r > — ao . 

f a ' -2d!- 1 / a - * = 2 sh- 1 /a ' ■ 



J „ v ( a - # • a — x ) * a 

Treating the second integral in the same way there are three 
columns to fill in, where the integral, in the notation above for 
12. is corrected for one limit or the other, and the factors made 
positive, so that the integral I should be real and positive. 

But all these forms are reducible to an integral of the first 

17 1 

— -, by taking ,—3 — = U as the variable ; thus 

m z f •(•-*.»' -b)du 



1 



(b- 


- u) \ (a — u . a - u) 
du 


1 


(b - uf 


l\ 


( l H l X + l ) 


'v 


\jb — u ' a — b b — u ' a — b) 
dU 



</{U + A . U + A') 



or an allied form. 



THE DYNAMICS OF MECHANICAL FLIGHT 




A 



i = i 






I I 



« II 

I S 











** 


1 


1 1 


i 


§ 


- 


S 53 




1 




> * r« H 








> 


' 


i J i ^ 










lr> 








- 






53 




^v 




1 " 


7 ? 


i 






1 


"<3 


^ s 


* 


{g 




§ l 


8 1 


1 






§ 


1 


1 1 


1 _ 


- i 




e 






- 




•~N 


S 


. ® 


§ 8 


. a"- 

" As 


A 5 


1 C * 

> s < 


1 1 


"«2 

1 


A 
A 




1 .3, 

s i 


1 


a \ « H 

1 II N 


« > 




i 7 7^ 


e 




I ° 


7 - ~5- 


it 




_ O -5 






£, 


J? — 


j* 


^>? II 

ll 




i 


"""^ s 


in ^ 

ii II 




ii 




i 





A. 
At 

A^ 

s " 



A a i | « 

_£ I 
> 5 ^ 



A 



e « - 



I II 



■Mil 



A 2,l T T T ? J! 



>>>> A 



> S 



S -^ i 



>> 



im cn cq 

II I II 



THEOEY OF A DISCONTINUOUS STREAM LINE 71 



A 
A 



1 




§ 


"0 


§ 


<s 




*"« 




e 


» 












o 


rO 


^ 


140 














o 








^~ 


& 






_ 



V5 >S 



9 

1 52 


53 § | § 


— u . a' 

ic) 

b . a — 


1 1 


i 


$, o i | 1 I 1 | 1 I 

i ° 7 7 




5 r& X 

3 CO C 


i o 



s 



53 4 
A * 

A 



«3 3 

^s o 



? I > *> :- 



A * 
A 



I I 

-ft 53 



I "l 



> > 



A^ 
A 



«i- rf ^ 



I I 






? 



> « 



I I 



I I 

<3 



« >>^ 



I 



8 
I 

A 
A 



53 




r^ 


<3 




"« 




rft 



y|>> 



7-2 THE DYNAMICS OF MECHANICAL FLIGHT 

The art of Integration of a function is to express the result 
as a new function of the constituents of the function, leaving 
them unaltered if possible ; thus 

(65) /'sec 6 clO = ch- 1 (sec 0) 

is preferable to log (sec 6 + tan 6) or log tan (J -n -\- \ 6). 

Professor Perry has shown the utility of the idea of lead and 
lag ; thus 

(66) I- C0S {nt + c) = n cos (nt + e + Jtt), 

v ' di sm v sin ' 

f C0S (7^ + e) dt = - C ° S (»* + e - 1 tt), 
J sin 7i sm 

a differentiation giving a /m^ of J77 to the phase angle nt + c, 
and a fo$ of J 77 being required in the integration. 



So also for the damped vibration, 

(67) ^e- mtC0S (nt + e) = y(m 2 + ™ 2 ) e~ mt cos (nt + e+Tr-tan 
v ' dt sm sm \ 



I 



mf cos / , , x 7 , sm \ m 

e~ mt . (nt + €)dt 



)S hit + c - tt + tan -1 - 



sm \/ [nr -f- 7i-) 



LECTUEE IV 

GYROSCOPIC ACTION, AND GENERAL DYNAMICAL PRINCIPLES 

A pew gyroscopic experiments were shown at the end of the 
last lecture, made with simple apparatus of bicycle wheels, spun 
by hand, either as an ordinary top (Fig. 41), or like a gyroscope, 
suspended by a stalk from a vertical axle (a bicycle hub) fastened 
in a bracket bolted to the under side of a beam, the beam resting 
on two step ladders for support (Fig. 42). 




Fig. 41. 

On this scale, visible to a large audience, spin sufficient can be 
given by hand, and no string is required ; so also with the 
Maxwell top (Fig. 43) twirled by a finger and thumb. 

The bicycle wheel can also be held by the stalk and brandished, 
to give the muscular sensation of its gyroscopic action when 
spinning, and so realise the influence on the steering of a flying 
machine of the rotation of the motor and screw. 



THE DYNAMICS OF MECHANICAL FLIGHT 



A bicycle wheel on its ball bearings makes an excellent 
pendulum for experimental illustration. After being tested for 
balance, and friction in producing a gradual decay of the revolu- 
tion, the wheel may be put out of balance by an iron bar between 
the spokes, and then it can swing as a pendulum through an arc 
small or large, and also make a complete revolution. The laws 
of pendulum oscillation, depending on the elliptic function, 
may thus be tested through an angle of oscillation however 
large. 




Fig. 42. 

To give the exact theory, with as few symbols and formulas as 
possible, of the steady motion of the gyroscope or top with the 
axle at a constant angle 6 with the vertical, a knowledge is 
presumed of the general dynamical principles of velocity and 
momentum, linear and angular, and their vector representation ; 
and use must be made of the dynamical lemma — 

" The vector velocity of the momentum, linear or angular, is 
equal to the vector of the impressed force or couple/' 

This is Newton's second law of motion, extended to the case of 
angular momentum ; and Maxwell's " Matter and Motion " may 
be consulted for an elementary demonstration of the essential 
principle. 

The vector representation of a couple, as in Statics, by its axis 
is thus required also, the couple due here to the preponderance, 
first moment, or leverage of gravity ; the notion, too, and 
determination of moment of inertia (M.I.) or second moment. 



GENEKAL DYNAMICAL PRINCIPLES 



75 



In the discussion of the gyroscope in Fig. 42 the MJ. is 
required about the supporting pin 0, and this is determined 
experimentally by letting the wheel swing in one plane like a 
pendulum, and measuring the length, I feet, of the thread of the 
simple equivalent pendulum which beats the same time ; denoting 
the M.I. by A, then 



(1) 



A 
Wh 



, A = Whi (lb-ft- 2 ), 



where Wh, the preponderance in ft-lb, is measured by hooking 
up the axle to the horizontal position 
by a spring balance, and taking the 
leverage as the product of the scale 
reading in lb, and the distance in feet 
of the hook from the pin ; having 
determined W in lb previously, by hang- 
ing the wheel from the spring balance, 
h is then the distance of its C.G from 0. 
So too we require C, the M.I. of the 
wheel about its axle ; and C can be 
determined experimentally, as shown in 
Fig. 44, with one of the bicycle wheels, 
by supporting the rim on a knife edge, 
and noting the length, V ft, of the equi- 
valent pendulum, and In! , the radius of 
the inside of the rim where the wheel is 
supported ; and then 




(2) 



I' = ti 



C 

Wh' 



c = wh' ty - h% lb-ft 2 . 



We call W, Wh, A, and C the physical constants of the wheel 
as a gyroscope or top, and if no spin is given, the wheel can 
make plane oscillations, like a single pendulum of length I ft, 
beating time, T seconds in small oscillation, where 



(3) 



A 

Wti 



ft; 



rasr VS B V: 



g Wti 



sees. 



76 THE DYNAMICS OF MECHANICAL FLIGHT 

Or, if precessing with angular velocity n (radians/second), in 
the same period 2 T, as a conical pendulum near the vertical, 



w »=£ 



A/f = x/"-?- l =% ^ f = TT^ (ft-lb). 



Take this wheel, weighing W lb, and spin it with angular 
velocity R (rad/sec), so that it makes ^- revolutions per second. 

The particles in a concentric ring, of radius r ft, have a 
velocity rR ; and if the density of the material is denoted by w, 
in lb/ft 3 , the energy of a particle, of volume dV ft 3 , is 



r 2 R 2 

(5) wdV — — , and of the 
2g 



wdV —- — = C — , ft-lb, 



where C = f rhcdV is the M.I. about the axle, in lb-ft 2 . 




Fig. 44. 



When the wheel is moved bodily with velocity v feet/second 

(f/s), the energy of translation is 

v 2 R 2 

W — (ft-lb), against C ~- for the energy of rotation. 



GENERAL DYNAMICAL PRINCIPLES 77 

And iust as W - is the linear momentum, in sec-lb, so C — 
J il 9 

is the angular momentum {A.M.) about the axle in sec-ft-lb. 

It is usual to take out W as a factor of C, dividing C into the 
factors W and k 2 , C = Wk 2 lb-f t 2 , so that k is a length in feet 
called the radius of gyration (Enler), or swing radius (Clifford). 

Familiar illustrations of M.I. are experienced in the muscular 
sensation of slamming a door, or brandishing a stick. 

Now spin this wheel with angular velocity R, rad/sec, giving 

7? 

A.M. C — , sec-ft-lb, and give the appropriate precession \x so 

that the axle will move at a constant angle 6 with the vertical 
downward. 

As R or ijl is increased, the axle can rise higher, pass the 
horizontal position, and when the beam in Fig. 42 stops the 
wheel we tarn to this other arrangement in Fig. 41, where the 
wheel spins like a top, with the point in a cup 0, and the axle 
OC at a constant angle 6 with the upward vertical. 

Selecting the gyroscope of Fig. 42 for discussion, as the motion 
is more under control, we draw the associated geometrical 
representations in Fig. 45, in which OC is taken to scale to 

represent the vector of C —> the A.M. about the axle. 

9 

Notice that we associate a vector of A.M., or of angular velocity, 
with a screw ; and we select the right-handed screw, so that the 
vector OC represents the direction of advance on the screw 
along the axle due to the rotation R. 

This involves spinning the wheel by a push with the left hand 
in Fig. 42, but with a pull in Fig. 41. 

The precession \x is then represented by a vector drawn 
vertically upward from 0, having a component \x cos 6 along CO 
in Fig. 45, and \x sin 6 along OA at right angles ; these are the 
components of angular velocity of the axle or stalk, but the 
wheel has an independent rotation round the stalk. 

The component \x cos 6 does not affect the wheel, only the stalk or 
axle, of which we ignore the inertia ; but the wheel rotates freely 
on the ball bearings of the axle with relative angular velocity 

R + \x cos 6. 



78 THE DYNAMICS OF MECHANICAL FLIGHT 

But it is the other component ju, sin 6 which carries the wheel 

round with the precession \x, and gives it the A.M. about OA, 

. u sin 6 
A - — . 

9 

The vector OK of resultant A.M. has the components OC, 

representing C — » and C'K, representing A > working 

with the gravitation units of the engineer, and keeping g carefully 
in its right place. 

The gravity couple is in the vertical plane COC, and its 
moment is 

(6) Wh sin 0, or A — sin 0, ft-lb, 

9 

represented by a right-handed screw vector draw towards us ; 
and this is to be equated to the vector velocity of K. 
Now the horizontal component of OK is 

(7) CK = OC sin + C'K cos 0, 
and the vector velocity of K is \x. CK, so that 

(8) Wh sin 6 = /* (C - sin + 4 ^ Sm(9 cos 

or dividing out sin 6 and #, 

(9) An 2 = CBp, + Afx 2 cos 0, 

the fundamental relation for steady motion, with the axle at a 
constant angle 6 with the downward vertical. 

But if, as in Figs. 41, 46, the angle 6 is measured from the 
upward vertical, the sign of cos 6 must be changed in (9), and 

(10) An 2 = CBfji - Afx 2 cos 0. 
Writing (10) 

/11X a CR ri 2 (CRY 2 (CB nY 2 ^ ( CRY 2 

(11) cos ° = a, " j? = (saJ ~ \Mn ~ J < (as) 

we notice that the top cannot reach the upright position if 
CR < 2 An, and the motion is then called weak. 



GENERAL DYNAMICAL PRINCIPLES 



79 



In a strong motion. CR > 2 An, and cos 6 can reach 1. 

after which the factor sin 6 = 0. 6 = must be taken., which was 
discarded from I 8 . 

In OC" make OL = I = ==r, the length of the equivalent pen- 
dulum for plane oscillation; draw AD at right angles to OL 
cutting OD at right angles to OK in D : and draw DL' hori- 
zontal to cut the vertical through in A'. Then 



,12 



and 



Oi/ _ = sin ODL _ gin ROC 

OL ~ sin OM ~ sin KOC" 

KC _ fi . KC _ A?z- sin _ » 2 

H" p. . KC Apr sin //- 

OZ = - . so that OL' =4 . 



and OL' is the height of a conical pendulum with precession f* 3 
keeping in the vertical plane COC" . 




Fig. 45. 
We can write 
(13) p . CA 



-i — sin #, 
o 



.1 " sin 
9 




Fig. 46. 



p . A" A = 

AIT = A l \ 



80 THE DYNAMICS OF MECHANICAL FLIGHT 

Multiplying, and dividing out ju, 

(11) CK . CK = U Vl sin 0\\ KM . KN = U - 

so that K lies on a hyperbola, with OC, OC for asymptotes. 

To draw the figure to a geometrical scale, to a length a as 

unit, we take 

B 



(15) 



OC_J_g 

a A n 



CK A ^ m0 ^ . ZM u 

— g — ~ sm 0, = — , 

a n n ft n 

A — sm 
C'K = 9 = KN _ n 

a j^n_ a ~ //,' 

9 

(16) ZM" . KN = ft 2 . 

But if this condition is not satisfied, the axle requires to be 
held at the angle 6 by means of a revolving gimbal frame COC, 
providing a couple, N ft-lb, in addition to the gravity couple 
Wh sin 0, and 

n_ CK 
sin ft 

- lW/isinfl. 



Wh sin 6 + JV = //. . 


UK , n 

. A -, 

ft a 


with /x = 


AT , 7l 2 CZ . C'it 

N = A — : — 

g ft'- sm 6 


• Wh sin (9 = 


/JOf . KB 



Thus X = again, if KM . KN = a 2 ; 
but if iOf . KN>a 2 , K lies on the concave side of the hyper- 
bola, and N is positive, holding the axle down ; and if released 
the axle would rise. 

An increase of the free precession /u would increase C'K and 
move K outside the hyperbola to the concave side, and the axle 
would rise, or, stated in Kelvin's words, 

" hurry the precession, and the top (or bicycle) rises." 

Arrest the precession by clamping the vertical spindle, and the 
axle falls and swings like a pendulum, having lost the directive 
gyroscopic effect ; but the spin of the wheel exercises a twisting, 



GENERAL DYNAMICAL PRINCIPLES 81 

which is felt as very strong when we try to hold the vertical 
spindle by hand. 

But the wheel in Eig. 42 can be supported by the finger at 
any angle, and no difference can be felt whether the wheel is at 
rest or spinning : not until the axle is allowed to fall. 

Harnraer the spinning top of Eig. 41, 42 with a stick, not too 
heavy, and the wheel flinches very slightly. 



It will be noticed that KC cuts the hyperbola again, in K\ on 
Eig. 46 ; and if it cuts the vertical in F, FKi = AY", and MN±, 
M t N are parallel to KK\ ; the precession 

(19) ft = JL 9^1 = n Ml, 

sin a a 



and is in the opposite direction in Fig. 45 and much larger than 
/x ; so that this motion is more violent, and hurrying the preces- 
sion will have the opposite effect, of making the axle fall, as we 
can show experimentally. 

Many attempts are being made still to utilise the gyroscope, 
for steering a constant course automatically, but any such action 
must be carried out through a light relay, as in the torpedo ; if 
the gyroscope is called on to do any work, it ceases to direct. 

Dismount the axle in Eig. 42 and hold it in the hands ; or 
else take the large wheel of Fig. 41, a 52-inch, and brandish it 
by the stalk, noticing the difference of muscular sensation 
according as the wheel is spun or not. 

With no spin, the wheel moves in the plane of the applied 
couple, so that the rotation is ab >ut the axis of the couple. 

But now spin the wheel, with angular velocity R, and A.M. 
j > 
C— , represented by the vector OC on the right-handed system. 

g r O 

If the wheel is swung upward \Uien holding the stalk out 
horizontal, an A.M. is communicated about an axis drawn to the 
right, and the axle OC swerves to the right, unless prevented 
by the muscular action of a couple represented by an axis drawn 
upward ; vice versa when the wheel is swung downward. 

M.F. G 



82 THE DYNAMICS OF MECHANICAL FLIGHT 

Swung to the J ■.%, k A.M. is communicated about an axis 

-, [downward I -, ,-, * (downward) -, ., 

drawn I upward j' and the axle swerves 1 upward | ; and xt 

requires to be held -, ul ^ [ by a couple whose vector axis is 

j j.1 I right i 

drawn to the ] ,%, f-. 
( left j 

The gyroscopic effect on the flying machine of the motor and 
screws has attracted attention. 

The biplane of Wright, Cody and Farman has two screws 
actuated in opposite direction by chains from the motor, one 
chain crossed, and here the gyroscopic effect is minimised. 

But the monoplanes of Bleriot and Santos Dumont have a single 
tractor screw in front, and when this is carried on a Gnome 
motor with cylinders revolving, the effect on the steering must 
be considerable, due to gyroscopic action. 

Putting the helm to port, as a sailor would say, sets up a. 
couple tending to turn the head to starboard, and its axis is 
downward ; so that with a right-handed screw the vector of A.M. 
receives a downward velocity, and the head of the machine 
descends. To ascend, the helm must be put to starboard. 

But if a turn is to be made to starboard, the axis of the couple 
must point to the right, and the helm of a horizontal rudder 
must be pushed down vertically, but held up if the head is to 
turn to port. 

With a left-handed screw on a Gnome motor, the gyroscopic 
influence is reversed ; a turn to the right will cause the 
head to rise and tail to drop. This sensation was dreaded 
by the pilot, and so he avoided the turn to the right as much 
as possible. 

The couple of reaction of the screw on the frame is a vector 
along an axis drawn to the rear, tending to make the machine 
bear heavier on the left wing, as if sailing on the port tack. 

Some such bias may help to improve the stability by giving a 
permanent bias to one side, in preference to the uncertainty of 
list, as of a crank tender ship. 



GENERAL DYNAMICAL PRINCIPLES 83 

The muscular sensation when the axle is brandished of a 
revolving wheel will illustrate the reaction on the bearings of 
the gyroscopic influence of the revolving machinery due to 
rolling and pitching of a steamer or flying machine. 

A paddle steamer has the main shaft across the ship, and so 
is affected gyroscopically by the rolling, not pitching. (Worthing- 
ton, Dynamics of Rotation.) 

But rolling does not affect the direction of a screw shaft, and 
pitching causes an extra reaction couple on the bearings acting 
across the ship. 

Suppose the steamer pitches through D° in T seconds ; the 
angular displacement 6 being given in radians by 

(20) 6 = " m sin^ - t I 

the angular velocity is given by 

/oi\ d6 t tt 2 D 

(21) Tt = f cos "" T' M " 3B0T' 

JD 

and if G — denotes the angular momentum in lb-ft-sec of the 

9 
revolving turbines, a couple must be supplied, with vertical axis, 
of maximum value 

(22) ° T -° 360^ fr 

= WW t^- . *£* ft-tons, 
360 60gT' 

at N revs/minute, with C = Wk 2 , ton-ft 2 ; and with bearings 
I ft apart, each carrying J IF, 

/oo\ Force on a bearing across the ship 

Dead weight on the bearing 

C ?Lt = ±n*Dk*N Dk*N 

= 9 ~ 360 x 60 x gTL ~ 5.600T2/ 

Working this out for I) = 3, N = 150, / = 30, T = 10, 
k = 3, the fraction is about ±^q. 



84 THE DYNAMICS OF MECHANICAL FLIGHT 

When the wheel is spun rapidly, so that OC is very much 
larger than C'K in Fig. 46, Ki is close to F, and \±i is large, 
giving a violent motion. 

But K is close to C, so that we may take 



(24) 



{%&) A - n sm ■== ix. — = /x— p sin 



c# 


OC . n CB . 


a 


: sm — -j — sm 

a A n 


Wh sin 6> _ 


. a GK 
n sm = a = u- 


An 


J 




An 2 





making /x small, and independent of ; as in the fundamental 
equation (9), when \j? is neglected. 

This relation for \l is true accurately when the axle of the top 
in steady motion is horizontal, and cos 6 = ; and the approxi- 
mation for any other angle is useful in popular elementary 
explanation of gyroscopic motion, such as given in Perry's 
Spinning Tops ; although the exact theory, as usual, is after all 
the simplest. 

The approximation in (25) amounts to assuming that the 
vector OK of resultant A.M. is undistinguishable from the axle 
OC, so that the velocity of C may be made equal to the couple 
vector, and it is employed freely in Worthington's Dynamics of 
Rotation. 

This is the case with the Earth, where the variation of latitude 
is insensible, and the approximation was employed by Poinsot in 
his treatment of Precession and Nutation (Connaissance des temps, 
1858). 

But the explanation still more popular will lead to an erroneous 
result, which strives to dispense with the idea of angular 
momentum, and works with angular velocity instead, as it 
would make 

(26) ix = - Rt instead of ^. 



GENERAL DYNAMICAL PRINCIPLES 85 

It is the angular velocity which is visible to the eye. and its 
vector as giving a line of instantaneous rest, shown by a coloured 
card on the axle of the Maxwell top ; but for complete dynamical 
treatment the A.M. is of greater importance. 

The most general motion of the axle would lead too far, where 
it makes nutations, and describes a path, either undulating or 
looped or cusped ; realised easily with the apparatus of Fig. 42. 
A condensed treatment will be found in Notes on Dynamics, 
p. 200, and here are two cases which lead to an algebraical 
solution of simple character. 

I. Hold the axle up horizontal, and, with no spin of the wheel, 
project the axle horizontally ; the motion is similar to a spherical 
pendulum. 

II. Spin the wheel and hold the axle up above the horizontal, 
so that when let fall it starts from a cusp and reaches the 
horizontal, and rises again to a cusp, and so continues in a 
succession. 

But the general case may prove of very complicated character. 

The lecture concludes with a digression on the simple princi- 
ples of Linear Dynamics ; the transcription should go on to a 
single sheet of paper, but it is all the dynamical theory required 
for a large number of familiar problems, such as those given in 
Notes on Dynamics (Wyman). 



86 THE DYNAMICS OF MECHANICAL FLIGHT 



Digression on Linear Dynamics. 

A wheeled carriage, electric tram, motor car, or motibus 
(Fig. 47), W tons, acquires velocity v f/s in t sees through s ft 
from rest, propelled by a constant force F tons. 

In a field of gravity, g f/s 2 , v is acquired in falling freely 

— seconds, through 7r - ft. 
9 % 

The car moves as if disturbed by a horizontal field, diluted to 

F 

W g > 



(27) Ft (or JFdt) = W- (sec-tons of momentum) 



(28) Fs (or jFds) = W V — (ft-tons of energy) 

s 1 

(29) = v (the average velocity in f/s) 
t A 



and Newton's Second Law of Motion is translated by equation (27). 
If break resistance B tons brings the car to rest in t' sees 
and s' ft, 

(30) Ft = Bt', Fs = Bs'. 

v 2 
Curves are drawn for W^-, v, and£; continued in a straight 

*9 

line for the middle part of a run at full speed v ; completed where 

the car comes to a stop. 

By giving the energy line a slope of F in W, it will represent 
the level of apparent gravity to a passenger, perpendicular to the 
plumb line. 

A sudden change in the plumb line will represent the jerk, 
as at stopping and starting. 

A passenger walking out at the front feels the floor sloping 
down and leans back ; the jerk of gravity restored tends to throw 



GENERAL DYNAMICAL PRINCIPLES 



87 



him on his back. Vice versa for leaving at the rear, also at 
starting, and the change from uniform velocity, as felt by the 
straphanger ; although this is smoothed down in practice, as in 
actual running the transition corners are smoothed down, and 
not so noticeable. 




Dead resistance may be allowed for by supposing the road 
slightly up hill at the angle of repose ; and no essential alteration 
is required, except in taking a little off F. 

What, for example, is the time gained by cutting out a 
station on the tube railway ? It is the halt and half the time of 
the stop) and start. 

And the time curve is the graph of a Bradshaw Time Tail:. 

No mention has been made of the acceleration of the car : the 
idea is difficult and it is proverbial the engine driver cannot 
grasp it. But the passenger feels it as a tangible sensation, 
especially on leaving the car at one end or the other. 

Acceleration appears, however, in g ; and we postulate the 
theorems that a falling body will acquire 

.2 

v f/s in - seconds, falling through — ft. 

And the proper place for g is below v and c' 2 : it must not go 
astray under W. 

The gravitation measure of force is used, suitable for dynamical 
questions in the field of gravity in which we live, and employed 
universally by the engineer. 



88 THE DYNAMICS OF MECHANICAL FLIGHT 

A second lecture, of one minute, would carry on with the 
Statics of the alteration of trim of the floor on the springs, due to 
passengers entering or leaving, and as the carriage is accelerated 
or retarded. 

The Law of the Spring is assumed as an experimental fact, 
based on Hooke's vague statement of the law — 

Vt tensio sic vis. 

A third lecture could be devoted to the simple pendulum, 
and the length required to beat time with the oscillation of a 
carriage body on the springs, vertical, pitching and rolling. 

Thus the vertical oscillation should synchronize with a pendu- 
lum of length equal to the set of the springs, the vertical distance 
the carriage body sinks down on superposition ; the law of the 
spring being supposed to hold. 

This is verified with a spring balance and a weight, a 32-lb 
shot, provided the scale can be graduated uniformly. 

But the logical and simple statement of the formula for the 
beat of the pendulum is 

(31) T=J l v noW^/i, 

if L denotes the pendulum length which beats the second; 
as it is L which is determined experimentally, and g is derived 
from it by the relation 

(32) g = «* L. 

The rolling and pitching oscillation of the carriage body on 
the springs would introduce the idea of angular inertia, with 
which this lecture began ; this is shown in its simplest form in 
the carriage wheels, in adding to the linear inertia of the carriage. 
The measurement of moment of inertia, or second moment, would 
run into a fourth lecture, provided we had the unlimited time 
at the disposal of Marchis in his Sorbonne lectures. 



LECTURE V 



THE SCREW PROPELLER 



There is no exact theory, it must be conceded, of universal 
acceptance for the screw propeller, and reliance is placed chiefly on 
an empirical factor based on experience and model experiment. 
employed in a formula which satisfies the condition of mechani- 
cal similitude, so as to predict from a small scale experiment 
the performance to be expected of the full size machine. 

A rational theory can be given of a hydraulic machine or tur- 
bine, when the water is compelled to follow a definite path : but 
where the fluid, air or water., is free to take its own course, as in 
the screw propeller, no exact treatment is possible until the stream 
lines have been determined. 

"Where the screw works in open water or air. the stream line is 
free to take a line of least action, and the shape is influenced 
to a great extent by the hull and fixtures in the neighbourhood, 
and the relative position of the propeller, effects which cannot be 
considered in a single formula. 

Numerous theories will be found in the Abstracts of the 
Report of th Aeronautical Committee due to various experi- 
menters, and one initial difficulty is to reconcile the conflicting 
notation employed by each writer ; it is time this notation was 
standardised. 

But the formulas are found to be in general agreement in 
making the thrust T proportional to 



90 THE DYNAMICS OF MECHANICAL FLIGHT 

1. The density of the fluid, w or ~, lb/ft 3 ; 

2. The disc area, S ft. 2 ; 

3. The square of the blade tip velocity, U f/s ; 

4. The slip s ; or more accurately to the product s (1— s). 
With an empirical factor / the formula for the thrust may 

then be written 

(1) T = fwS g s (1 - s), lb, 

and this is the weight of a cylindrical column of the fluid, of 
cross section S, and height 



(2) /|S(l-«),or/«(l-*)of ff = -g. 

The formula agrees then in making T = when s = and 
there is no slip, and the screw, of uniform pitch, advances in the 
fluid as if in a solid nut ; and also when s = 1, and there is no 
advance, and the screw cuts a hole in the fluid and swirls the 
fluid round. 

But with a slip s between and 1, the reaction of the fluid is 
against the rear of a blade, and a thrust is obtained. 

A negative value of s would imply that the screw was being 
turned by the stream through it, as a windmill or turbine. 

Working on the sails of a windmill or ship the wind strikes the 
rear of a sail and urges it forward, as in Fig. 48. 

If IF represents IF, the true wind over the water, and OV the 
velocity V of the ship through the water, then VW represents the 
apparent wind Q, as felt sweeping across the deck and filling the 
sail ; and the direction of TIF is given by the vane on a mast, or 
smoke from a chimney. 

On the Newton theory the thrust on sail area A ft 2 is given by 

(3) T = A (i^l! = A ( Q ^ " J, lb, 



THE SCREW PEOPELLEE 



91 



as we have taken previously when Q is given in f/s ; but when 

the speed is given by K in knots of 100 ft/minute, 

fA \ mi (K sin a N 

( 4 ) 



A 



12 



ib, 



because 60 knots is 100 f/s, 12 knots = 20 f/s. 




Fig. 48. 

On the diagram of velocity of Fig. 48, VWV = a, 
(5) via = Ow — Ov, Q sin a = W sin /3 — V sin 6, 

and the propulsive force in the line of the keel is 

IF sin /3 - Fsin 0\ 2 



(6) 



T sin 6 = A sin 



12 



if W and V are measured in knots. 

One H.P. of 33,000 ft-lb / min. is 330 knot-pounds, so that 
A V sin 6 I W sin /? - V sin 



(7) 



the sail H.P. = 



330 V 12 

Working this out for a ship of the size of the Preussen, spreading 
A = 40,000 ft 2 of canvas, with 6 — 30°, a = 60°, Q = 12 (18) knots, 
V— 10 (15) knots, the sail H.P. is then about 450 (1,534). 

An ice boat would run like a windmill unloaded ; and at full 
speed OV, the vane on the mast would point parallel to the sail. 

For the screw propeller, as far as theory can go at present, 
we begin with Rankine's treatment in the Transactions of ^the 
Institution of Naval Architects, 1865, following his notation as 
closely as possible. 

A screw surface, of a true screw, is swept out by a straight 
line intersecting an axis at right angles ; and the line advances 
along the axis and turns at the same time in a constant ratio ; 
and the advance for a complete revolution is called the pitch, 
and denoted by p, and measured in feet. 



92 THE DYNAMICS OF MECHANICAL FLIGHT 




Fig. 49. 



THE SCREW PROPELLER 93 

Engaging is a fixed nut, the pitch p is the axial advance for a 
complete revolution of the screw. 

A co-axial cylinder will intersect the screw surface in a uniform 
helix, shown by a straight edge of a flat piece of paper, after 
the paper is wrapped on a cylinder ; also by a thread or tape 
wound on the cylinder. 

When the paper is flattened out again, the helix appears as a 
straight line OC, and rolling the cylinder on the paper makes 
the helix roll on the straight line. 

Draw OD parallel to the axis of length p ; then DC at right 
angles to OD is %rr the circumference of the cylinder, for a 
radius r feet. 

The section of the screw blade on the helix of circumference 
DC, limited by two planes perpendicular to the axis, is shown in 
plan by A A' ', and by AB in end elevation ; and so for any other 
circumference, DC or DC\, by OA C A ', or OA1P1AJ (Fig. 49). 

Consider the motion of the fluid relatively to the screw, with 
the fluid approaching with the axial velocity u f/s,and the screw 
making n revs/sec. 

Draw CR in DC produced, representing to scale '2-rn, the 
velocity of the point C on the screw ; and draw CD representing 
the velocity u to the same scale ; then draw RV parallel to CO ; 

denote the angle OCD by 0, so that tan = v~ ; and then 

CV will represent the velocity pn, which Rankine denotes by v' ; 
and he puts v' = p r n, so that Rankine's p' is our p, and there is 
no need for an accent. 

Then RU represents the relative velocity of approach of the 
fluid with respect to the blade at C, and URV is called the 
angle of attack, and denoted by a, so as to correspond with 
KirchhofFs diagram (Fig. 4). 

Rankine assumes that the fluid is deflected by the blade element 
AA', so as to stream past it in the direction RV with relative 
velocity RQ, where UQ is drawn perpendicular to RV ; the com- 
ponent QU being used up in producing the thrust on AA'. 

With respect to the frame which holds the screw, the fluid 
streams away with velocity represented by CQ, having the axial 
component CV, denoted by r, and a transverse component VQ. 



94 THE DYNAMICS OF MECHANICAL FLIGHT 

The velocity np — u, represented by W, is called the slip 
velocity; and its ratio to pn is called the slip or slip ratio of the 
screw, and denoted by s ; so that 

(8) s = PlL^l = 1 -Ji , u = pn (1 - s), 

pn pn 

and then 

(9) v - u = UV = UQ cos = UV cos 2 = (pn - u) cos 2 0, 

or in Rankine's notation 

(10) v — it = (v' — u) cos 2 8, v = up (1 — s cos 2 6) ; 

a result written down by Rankine as if it was obvious, and 
required no explanation. 

Relatively to a blade element A A' of the screw, the fluid 
approaches on a spiral of angle 6 — a, and pitch 

(11) 2tt r tan (0 - a) = - , 

n 

and the fluid is leaving the screw on a spiral of pitch p with 
respect to the blade ; but with respect to the frame, the spiral 
has an angle CQV, and a pitch 

(12) 2tt r tan CQV = 2tt r tan 6 — = p — = * 1 ~ s c f S - 

v ; ^ VV £ pn - v 1 s cos 2 6 ' 

Draw AA" parallel to UR ; then if A A" does not cut the 
development A 2 A 2 ' of the preceding blade of the screw, and if 
the fluid is supposed to behave like a dust cloud of non- 
interfering particles, the fluid crossing A"A 2 ' will pass through 
the screw undisturbed by a blade ; and it will form a wake 
like the helical strand of a rope, stationary with respect to the 

surrounding fluid and of pitch — ; but moving past the frame 

with velocity u and making n revs/sec. 

There is a similar strand proceeding from the rear of a blade, 
from which all the dust particles must be supposed swept out. 

But the particles which strike the blade element, of length 
AA', and breadth dr, are measured, in lb/sec, by 



THE SCREW PROPELLER 95 



(13) 10 u .A' 'A' dr = pn r , u AB dr, 

where A "A' is the parallel to CE, and 

(14) w = 1 is the density in lb ft 3 , C = - is the S. V. in ft 3 /lb ; 

since 



(15) 



.15 ca 



If, however, AA" cuts the preceding blade, so that the blades 
interfere and screen each other in succession, then no fluid 
particles escape being received and deflected ; so that the fluid 
acted upon is measured by 

(16) * 2- r dr = l ± dS, lb sec, 

c c 

and this is contained by the cylindrical sheet of thickness d r 
and mean radius r, and cross section dS = '2-rdr. 

On these assumptions a blade leaves a vacuum in its rear ; 
but the current of fluid which encounters the face of a blade 
element A A' is driven off in a stream, parallel to the blade in 
the motion relative to the screw, but with respect to the frame 
in the direction CQ. 

If unchecked by the surrounding fluid, the stream would 
continue in a straight line, by what is called sometimes the 
centrifugal tendency, really the First Law of Motion ; and the 
wake current would grow in diameter in a conical or trumpet 
shape. 

^Vith a screw working in water such a state of centrifugal 
motion would be impossible unless the water was shattered into 
drops ; but the liquid particles are compelled by the surrounding 
water to describe spiral lines, of angle 9', and pitch 

1 — s cos 2 # N 



V 



s cos 2 6 



and the centrifugal effect must be balanced by a pressure 
gradient, in a radial direction. 



96 THE DYNAMICS OF MECHANICAL FLIGHT 

With no such cavitation of the water past the screw, the 
principle of the equation of continuity in hydrodynamics must 
he introduced ; but this principle is ignored when the fluid is 
treated as above by Rankine, as if it was a dust cloud of 
non-interfering particles, as in the Kinetic Theory of a Gas. 

So far the theorems are geometrical ; a dynamical principle 
employed by Rankine asserts that the forward thrust P of the 
screw, in lb, is given by the axial backward momentum in 
sec-lb, communicated per second. 

The increase of axial velocity is from (9) 

(17) v — u — (pn — u) cos 2 6, 
so that in the first case of blades not interfering, 

(18) dP x = &^ AB dr v -^^ = ill 1 - u ? cos 2 AB dr. 

C a Cg 

Denoting by j\ the fraction of the disc area occupied by the 
blade area projected on the plane of the disc area perpendicular 
to the axis of the screw, so that AB = 27r/ir, 

9) dP 1 = { lm ~ g nr ' (1 - sin" 6) Zirfrdr 



(P'i ~ «)' r (.y- rdr - ^ rdr \ 
Cg J1 \. p- + <UrW 



w 7 here dS denotes the ring element of the disc area S ; and 
integrating, with p 2 + 4tt 2 r 2 = OC 2 , 

(20) P l = iS^f 1 {S-^lo g ^ = (?^jVf }S lI,' 

suppose, where 

(21) M = l-^lo g §g 

1 ° OP* °° OP? 

= 1 - & ' oc * _ oci 

O'P 1 OP- 



THE SCREW PROPELLER 97 

If D denotes the extreme diameter of the screw, D 1 of the 
boss, and U the tip velocity due to the screw revolving n times 
a second, U = ^ Dn, f/s ; and then 

(22) P 1= ^L S %SM 

{ 

p2 ™ s % br (^ 2 - A 2 ) m 



Cg t^D^ 



-•SO-*") g'/* '"■ 



But in the second case when the blades begin to interfere, 

(23) dP 2 = — ab = Kl „ ; cos 2 <9 . 2-nrdr 

9 Cg 

_ u(pn — u ) ( to _ 2irp 2 rclr 

~~Cg V ' V' 2 + 4ttV 

(24) P 2 = ^ " tt > SM 

Cg 

= tf S (l- s)SM 

Cg 



w 



£L(l-mE!s(l-s)M; 



2tt V D' 2 )Zg 



this is the formula of Coriolis, agreeing in shape with the 
empirical formula (1). 

For a screw working under water without cavitation, where 
the axial velocity increases from u to v in passing through the 
screw, the principle of continuity requires the cross section of 
the stream to diminish, inversely as the axial velocity. 

A radial current motion must then exist inside the screw 
stream ; and in the formula of Coriolis the stream wake leaving 
the screw would be of reduced diameter. 

Rankine, from observation of the screw propeller, prefers to 
take the screw wake as of full screw diameter, make wvdS, 
lb/sec, the quantity of water acted on by the element A A', and 
so giving an element of thrust 

M.F. H 



98 THE DYNAMICS OF MECHANICAL FLIGHT 

(25) dP, = wvdS v ~ u = id v & n " u) cosWdS 

9 9 

(26) dP 3 - dP 2 = w fe ~ u) ^ n ~ g> ooMS 

= u; (g? ~ ^ sin 2 cosWS 
9 

= iv P— s 2 sin 2 <9 cosWdS. 
9 

Eankine puts 

(27) ^ = coi6= r = n 

p Ait 

dS = Zirrdr = £ qdq, sin 2 = X - , cos 2 



l + g 2 ' " " i + g 2 ' 

t? Ll + 2 2 (1 + <Z 2 ) 2 J 

giving a slight increase in P 3 over the P 2 of Coriolis, depending 
on s' 2 the square of the slip ratio. 

In a numerical application the calculation is usually close 
enough when the area is ignored of the boss of the screw, so that 

(30) i\ = 0, 6 X = |7r, q x = 0, S = 7rr 2 = t- cot 2 o , 

47T 

(3i) m = 1 - Jegggggg 

v y cot 2 o 

(32) P 3 - P 2 = * -g^. .*? 6- 2 (log cosec W Q + cos 2 o ). 

A knowledge of the tip velocity U is required in a calculation 
of the strength of a screw blade, especially with the high velocity 
required with a wooden propeller of a flying machine. 



THE SCEEW PROPELLER 99 

On a diameter D feet, the ratio of centrifugal force (C.F.) to 
gravity g is given by 

(33) G ' F ' = U2 

gravity \Bg 

will give an idea of the stress in the material. 

If d is the diameter of the circle described by the C.G. of a 
blade of weight W, the centrifugal pull at the root of the blade 

idg ^kT u 2g 

For instance, in a screw 7 ft in diameter, 22 ft in circum- 
ference, making 1,200 revs per minute, or 20 revs per second, the 
tip velocity U = 22 X 20 = 440 f/s, which is 300 miles an 
hour, and at the tip 

C^ (440)1 =1,700 about. 

v ; g 7 x 16 

If the blade weighs 20 lbs and its C.G. describes a circle of 
3*5 ft diameter, the pull at the root of the blade is 

(36) 20 x | X (4 |^ = 17,280 lbs, nearly 8 tons. 

The tension length in a ring of metal on the circumference at 
this speed U f/s is 

(37) ^ = ^i 2 = 6,050 ft ; 

and for steel of density 500 lb/ft 3 , this would imply a tension of 
about 9 tons/inch 2 . 

For a smooth screw, the axial thrust dP implies a circum- 
ferential thrust tan 6 dP and so a turning couple, in ft-lb, 

(38) dL = r tan 6 dP = £. dP 



(*» §-h i-h **-& 



H 2 



100 THE DYNAMICS OF MECHANICAL FLIGHT 

and this is evident from the principle of Virtual Velocity or 
Work ; because 2tt L is the work in ft-lb done by the couple 
L ft-lb in a complete revolution of 2ir radians, and this work 
with the smooth screw is equal to pP, the work done by the 
thrust P lb pushing through the pitch p feet. 
The shaft horse power (S.H.P.) at n rev/sec is 

(40) BJLP.-^-lE. 

while the thrust horse power 

(41) T.H.P. = gi, 



and the propeller efficiency e of the screw is measured by 

T.H.P. __ ii_ 
S.H.P. ~ pn 



,, m T.H.P. u ., 

( 42 ) e = trwrr = z= = 1 



or 

(43) efficiency + slip = 1. 

Thus the 

If the screw is run as a windmill or turbine, in air or water, so 
as to take up energy out of the current, np is less than u, and the 
angle of attack changes to the other side of the blade AA'. 

For a given u and by variation of n or np, the S.H.P. 
transmitted is a maximum when np = J u in the first case, 
but np = J u in the second case where the blades interfere. 

The old-fashioned windmill with four narrow sails (Fig. 50) 
will belong to the first class, and will be doing most work when 
run at one-third the unloaded speed ; and then its 



THE SCREW PROPELLER 



101 



But the new Canadian form of windmill (Fig. 51), like a smoke 
jack, with numerous vanes which screen each other, should run 
at half the unloaded speed, so as to develop maximum power, 
and then the 



(46) 



S.H.P. 



1 WIL 2, 



4 550g 



SM. 



These theories should be capable of experimental verification 
on a large scale. 





Fig. 51. 

The theory of the windmill and air propeller should be the 
same ; but in the treatment above, no notice has been taken of 
the fluid after it has passed the screw. 

The equation of continuity of fluid motion is ignored in the 
turbulent motion of the wake, and the fluid is assumed to 
behave as a cloud of dust, so that two currents can pass through 
each other without interference. 

But with a screw under water, where the backward stream 
leaves the screw with axial velocity v, the stream line must make 
a spiral of pitch p with respect to the revolving screw, but with 

pitch p with respect to the frame. 

Also with v greater than u, an inward radial velocity is 
required to prevent cavitation ; the current acted on by the 
screw is of greater area ahead, and the stream lines converge 
towards the axis in passing through the screw (Fig. 52). 

But no theory so far has been able to assign the amount of this 
convergence for a screw working in unlimited water. 



102 THE DYNAMICS OF MECHANICAL FLIGHT 

Let us examine an ideal case, in which the screw is working in 
a cylindrical tunnel which prevents radial motion in the water, 
and then continuity and absence of cavitation requires u = r, for 
the water running full bore. 



\t 



A r 



Fig. 52. 



There is now no generation of momentum sternward, and the 
preceding treatment would imply a zero thrust. 

The thrust is due, however, to a change of pressure, and the 
screw acts as a turbine pump. 

The water feeds the screw with a uniform stream velocity u, 
and leaves with the same axial velocity u, but with a transverse 
velocity 



(47) 



O y 

(pn — u) cot = (jpn — u) ; 



that is, the angular velocity round the axis is 

n II 

2 it (n ), or the water is making n , revs/sec. 

P p 

The screw now spins a rope of the wake, at the rate of angular 
momentum, in lb-ft 2 /sec, 



(48) 



to S k 2 u : 



t'i), 



1 S 

or, ignoring the boss, and putting k 2 = -D 2 = -x- 



(49) 

giving a thrust 



L = 



(*-f)* 



THE SCKEW PEOPELLEE 



103 



(50) 



2ttL 



(-}) 



2ttS- 2 ; 



or, expressed in terms of the tip velocity U, and slip s, 
(51) P 4 = w |l {^)^ (s - s") = v 5 g-( S - a"), 

agreeing with formula (1) with / = 1, and assuming that the 
screw wake receives a uniform swirl. 




Fig. 58. 



In a paddle steamer p should represent the circumference of 
the pitch circle through the centre of the floats, S the area of a 
float, and 



(52) 



T=fwSu np ~ u =fwS — s (1 - s). 

g 9 ' 



The vessel advances as if a circumference of length — engaged 
in a horizontal rack, to give it a velocity u (Fig. 53). 
In sea water, of specific volume 35 ft 3 /ton or 



35 



2,240 



64 



1 ll — o 



ft 3 /lb, and with g — 32, Cg 

And if the thrust P 4 is given in tons, 

Cq = 35 X 32 = 1,120. 



104 THE DYNAMICS OF MECHANICAL FLIGHT 

Thus for one of the four screws of the Mauretania, 16 ft in 
diameter, 18 ft pitch, making 150 revs/min with 7 per cent, slip, 

n = — - = — , pn = 45 f/s, about 27 knots ; 

bO A 

u — pi (1 — s) = 41 • 8 f/s, say 25 knots ; 
s = 0-07, 1 - s = 0-93, S = 64 tt 2 ft 2 ; 

we find Pi is about 90 tons, so that at 42,000 tons displacement, 
the resistance of the water is equivalent to an incline of 1 in 
42,000 -r- 360, say 120, a resistance of about 20 lb/ton, reckoned as 
on a railway ; and the S.H.P. works out to about 66,000. 

These numbers have been manoeuvred so as to agree with 
practical results by the choice of a very small slip, 7 per cent. ; 
an estimate of 15 per cent, would be more likely, but this would 
double the value of P 4 , so that we should require to take / = J 
to bring the former into agreement, and now the speed would 

have sunk to 45 (l —-^L) = 38*25 f/s, or 23 knots. 

A provisional estimate of f may be made by taking it as the 
fraction of disc area made by the projected blade area. 

Tested on an air propeller, two-bladed, with diameter D = 15 ft, 
and making 450 revs/min with a slip of 36 per cent., and giving 
a thrust of 1,000 lbs, we should find that we should have to 

take/ =— ' implying that the blades were sections of 10° in the 
lo 

disc area projection. 

The thrust P 4 in (51) produces an increase of pressure 
(53) ^=w^s(l-s)lb/it\ 

r 2 

or a head of-^— s (1 — s) ft; 

and this is the height to which water con Id be driven, using the 
screw enclosed in a tube as a centrifugal pump. 



THE SCREW PROPELLER 105 

A numerical comparison can be made with a four-stage 
centrifugal pump, described in the Engineer, June 17, 1910, 
which pumps 110 gallons/minute to a head of 225 feet, at 1,440 
revs/min, and B.H.P. 10. 

The loss of energy, measured in ft-lb/sec, is 

(54) 2tt Ln — P 4 u ; 

and of this, half is thrown away in the rotating energy of the 
wake ; the other half is lost by shock on the blade. 

This last energy can be recovered if the leading edge A' of the 
blade is given a zero angle of attack, so that the screw has 

a gaining pitch, from —at A' through its mean value p to some 

final pitch j/, and the blade A A' is cambered. 

If the camber is parabolic, so that cot 6 increases uniformly in 

the axial direction, the mean effective pitch p is the harmonic 

u 
mean of the initial pitch — and final pitch p' , 



(55) !_=l(!L + i 

v 7 n 2 \u p 

The system of gaining pitch is always adopted with a turbine, 
intended to run at a given speed n in a given current u ; the 
guide blades may be taken as the equivalent of another screw 
fixed in front. 



Two screws on the same shaft line were employed in one of 
the earliest screw steamers, so as to recover the rotational 
energy of the wake ; the system has been brought forward 
lately by Colonel Rota, of the Italian Navy ; the system seems 
applicable to the Gnome motor on a flying machine, when the shaft 
is made to carry a screw as well as the cylinders, and is allowed to 
revolve in the opposite direction. 

In this way the revolutions of each screw are halved, while the 
relative motion of the axle and cylinder remains the same as is 
desirable in practice. 



106 THE DYNAMICS OF MECHANICAL FLIGHT 

So long as the slip s is small, the energy recovered from the 
wake would not be worth the extra weight and complication of a 
second screw. 

But with the two screws the slip s may be made as large as 
50 per cent., s = J, and then each screw is pulling hardest for its 
weight, so that size can be reduced and weight economised. 

With uniform pitch there would be no economy of efficiency, as 
the energy recovered from the wake is lost again in shock at the 
second screw ; but with appropriate gaining pitch the theoretical 
efficiency can be made perfect. 

Large slip is preferred at sea for driving against a head sea, 
and diminution of racing. 

Eacing of the screw is due chiefly to variation of axial flow ; 
the variation of the longitudinal velocity of the water in wave 
motion has more influence than the accompanying vertical 
component. 

With a fine pitch and small slip this velocity variation causes 
a rapid change in s and L, not so rapid when s is large. 



LECTURE VI 



PNEUMATICAL PKINCIPLES OF AN AIR SHIP 



The flying machine as a practical success is only some 
two or three years old ; but it looks as if it will displace the 
air ship balloon, with a large gas bag to give the ascensional 
force. 

The air ship lighter than air is, however, still on its trial, 
and so we proceed to discuss the pneumatical theory involved ; 
for the detailed calculation a reference must be made to 
Chapter VIII. of my Hydrostatics. 

The first practical balloon to make an ascent with a man 
dates from 1783, the hot air balloon of Montgolfier (Fig. 54). 

The legend goes that as Madame Montgolfier's silk dress 
was airing before a fire, it became inflated and rose to the 
ceiling. 

Montgolfier followed up the idea, on a small scale at first, 
on the impression that the hot air was some new kind of gas ; 
and finally, as a paper manufacturer, he was able to make a 
fire balloon large enough to take up the first two real aeronauts, 
Pilatre de Rozier and the Marquis d'Arlandes, in November, 
1783, from the Chateau de la Muette in Paris, and so realise 
finally the dream of the poet and artist of antiquity. 

The principle is seen in the ordinary toy hot air balloon ; 
the air in the balloon is rarefied by heat to an extent such as to 
make the total weight of the balloon, car, and passengers, and of 
the hot air it contains, equal to or less than the weight of the 
external cold air displaced. 



108 THE DYNAMICS OF MECHANICAL FLIGHT 

The experimental laws of pneumatics required in the theory are 
embodied in the gas equation 



(1) 



e 



lhv* 



connecting the pressure p, lb/ft 2 , specific volume r, ft 3 /lb, 
and absolute temperature 6, which we take Centigrade, with 
p2, v 2 , 2 , in another state of the same given quantity of a gas. 




Fig. 5i. 



Fig. 



Fig. 56. 



This equation expresses Boyle's law when the temperature 
is constant, and the law of Charles, when 6 varies, and either p 
or v, one at a time. 



Denote by W, lb, the weight of the balloon, car, and 
aeronauts, corrected for buoyancy of the air, as if weighed in a 
vacuum, and denote by W lb the weight of atmospheric air 
they displace, so that W — W lb is the apparent weight when 
weighed in air ; denote also by V, ft 3 , the capacity of the 
gas bag of the balloon, so that M = Vp lb denotes the weight 
of atmospheric air which fills the balloon, at a density p, lb/ft. 3 

When the air inside is raised in temperature from to 0' 
degrees, absolute Centigrade, part of the air will flow out of the 
balloon, leaving the rest at the same pressure, p, but at density 



PNEUMATICAL PRINCIPLES OF AN AIR SHIP 109 

(2) p — , and therefore of weight Vp — , = M _,, lb. 

6 

The balloon will be floating in equilibrium when the weight 
of the balloon and hot air it contains is equal to the weight 
of surrounding cold air displaced ; that is when 

(3) W + M -, = W + M, 


21 - W + W & - = W - W 
0' M ' '' M - W + TV" 

determining 6' — ^. the increase of temperature required to 
rise. 

The balloon is now in unstable equilibrium, like a bubble 
of air compressed to the density of the surrounding water ; and 
it will begin to rise, as it cannot descend. 

The balloon will continue to rise and the hot air to escape till 
another stratum is reached, at height z ft, suppose, where 
the density is p M and absolute temperature d, ; and then the 
pressure p z is given by the gas equation (1) 



(4) Vz = P — -r = - ^ ■ 



Reference must be made to Hydrostatics, Chapter VIII., for 
the further theory of Montgolfier's hot air balloon ; but the 
principle was soon abandoned in favour of the hydrogen balloon, 
invented by the chemist Charles a few months later, having an 
advantage that the lift can be obtained with a gas bag much 
smaller (Fig. 55). 

This is of great importance for military use, where the balloon 
is tethered, and size must be kept down on account of the wind ; 
also in the large air ship, intending to take up numerous 
passengers, and to keep up in the air and make a journey as 
long as possible. 



110 THE DYNAMICS OF MECHANICAL FLIGHT 

For economy the ordinary balloon of sport is filled with 
coal gas from the neighbouring works ; and on the aeronauts 
rule that 1,000 ft 3 of this gas will lift about 40 lb, while 

1,000 ft 3 of air, at 12-5 ft 3 /lb, weighs 1,000^-12-5 = 80 lb ; 
this makes the S.Y. of this gas double that of air, or 25 ft 3 /lb. 

Generally 1,000 ft 3 of a gas of S.V. n fold of the air will lift 
P lb, given by 

/e\ on 80 80 

(5) p = 80 n = 



n 80 - p 

represented by a hyperbolic graph. 

To rarefy the air in a Montgolfier fire balloon to double S.Y. 
would require the temperature to be raised from to 273° C, 
from 32 to 520° F. ; but this heat would disintegrate a fabric like 
paper or silk ; and if replaced by asbestos the weight of the skin 
becomes excessive. 

With pure hydrogen we may take n as large as 14, so that 
1 ton of hydrogen can lift itself and 13 tons more in the air ; and 
as 1 ton of air bulks 2,240 X 125 = 28,000 ft 3 , 1 ton of 
hydrogen bulks 28,000 X 14 = 392,000 ft 3 ; and 2 tons of 
hydrogen, or 784,000 ft 3 could lift 26 tons ; this is the volume 
of a cylinder 500 ft long and about 50 ft diameter, something 
like the Admiralty air ship. 

Suppose it is required to weigh a ton of hydrogen in the scales 
and at atmospheric pressure, not compressed to 100 atmospheres 
in a steel cylinder, 13 ton weights would be required, and placed 
in the same scale. 

But compressed in a cylinder to 100, or generally to x atmo- 

x x 

spheres, to -of air density, the ton of hydrogen would displace - 

of a ton of air, and would weigh an empty cylinder in the other 

scale and 1 — - tons. 
x 

n 
Thus with n = 14, x = 100, 1 = 0*86, at a volume 

3,920 ft 3 , say a cylinder 8 ft in diameter and nearly 80 ft long; 
or 560 cylinders of a usual size, 9 ft long and 1 ft diameter inside. 



PNEUMATICAL PRINCIPLES OF AN AIR SHIP 111 

And the price, too, is an important consideration, nearly £1,000 
for a full-size air ship ; although it is claimed that hydrogen 
can be produced at Id. per metre 3 or 35 ft 3 per penny, so that 
800,000 ft 3 would cost about £100. 

Suppose the gas employed for inflating the balloon is n times 

lighter than the air, of density — lb/ft, 3 and S.Y. nv ft 3 /lb, 

( — and nC in the previous notation] ; and let U ft 3 of 

this gas, weighing P = — lb, be allowed to flow into the 

nv 



balloon. 

The balloon 


will be 


on 


the point 


of rising 


when 








(6) W + P = 


W + nP, 


p- w - 

n - 


- W 

- 1 ' 


U 


= (W 


- W) . 


nv 
i — 


1 


(7) U : 


W - 
A 


W 


, where A = 


1 

V 


1 
nv 


= p - 


p 

> 

11 







and A is the lift of the gas in lb/ft 3 , 1,000 A in lb/1,000 ft. 3 

The balloon, like the bubble compressed in water, is in 
unstable equilibrium, and will begin to rise ; and to carry the 
balloon clear of neighbouring obstacles rapidly, it is advisable 
that the volume U or weight P of gas should be increased, to 
give an ascensional lift, which at starting will be a force 

(8) (n - 1) P - (IF - W), lb. 

As the balloon rises, the gas contained in it will expand until 
the envelope is completely inflated, and the gas will now occupy 

V ft 3 ; this will take place where the density of the air is — T p, 

and - — the density of the gas, the temperature of the gas 

being supposed unaltered. 

The ascensional lift force will now be 



(9) (n-l)P- (V- Wfy 



II) 



or W ( 1 — ) lb less than at starting, 



112 THE DYNAMICS OF MECHANICAL FLIGHT 

The balloon will still continue rising, but now it is very 
important that the neck of the balloon, should be left open, 
to allow gas to escape as the balloon rises into the more rarefied 
air, and so equalise the pressure of the interior gas and surround- 
ing air ; otherwise the pressure of the gas, if imprisoned, might 
burst the balloon, as Charles found when he started experi- 
menting with small hydrogen balloons, sealed up. 

At the height z ft, where the density of the air is p z , the 
ascensional force, in lb, will be 



(10) V Pi 


■(•■ 


i_ 

n 


-w - ■ 


W'PA 












= [(n- 


-1) Q + W 


" 1 ?j - 
P 


- W 








= (VA 


+ w) e* - 

p 


- W, 




on putting 














(11) 


Q = 


Vf- 


_ M 


M - Q = 


(n - 


1)'< 



VA 



where Q denotes the weight of gas, and M of air, of volume V 
which would fill the balloon on the ground. 

The lift is zero and the balloon comes to rest where 

(12) P± = h 6 - W 

{ J P p ' 6 S VA + TF" 

and in an isothermal atmosphere, 

no\ P» Pz - z - ? i P 7i VA + W 

(13) ^ = *• = e k , z = k log !L = fjc log 10 ^, — , 

P V Pz w 

where k denotes the height of the homogeneous atmosphere, say 
28,000 ft, with an atmospheric pressure on the ground of one 
ton/ft 2 ; and fi == 2*3, the modulus of the natural logarithm. 

Here again the equilibrium is unstable ; as if the balloon 
rises a little more, it loses gas through the neck as well as 
by diffusion ; and if it descends the balloon is bulged in and 
loses buoyancy displacement, and the pilot recognises this at 
once by the crackling and pucker of the skin. 



PNEUMATICAL PEINCIPLES OF AN AIE SHIP 113 

A free balloon is either rising or falling, and it must be steered 
in a vertical plane, either by throwing out ballast or letting 
off gas by the top valve ; but it can be kept at a moderate 
average height by a rope trailing on the ground, or, in the case 
of the American Wellman air ship, by the buoy trailing on the 
sea. But here the violence of this equilibrator among the waves 
was the cause of failure. 

The calculation is given in Hydrostatics, § 241, Chapter VIII., 
of the effect of throwing out ballast. 

Pilatre de Eozier took up the mad scheme of combining the 
two systems, Montgolfiere and Charliere, of Montgolfier and 
Charles, into one balloon, his Charlo-Montgolflere, as shown in 
Fig. 56, a spherical gas bag of hydrogen, on the top of a vertical 
cylinder to carry heated air. 

His idea was to preserve the hydrogen sealed up, and to use 
the hot air as a regulator for rising and descending without 
carrying ballast, and so keep the air an indefinite time. 

The inevitable came to pass very soon, in about half an hour 
from the start from Boulogne, June, 1785. The gas bag burst 
as soon as a moderate height was attained, about 5,000 ft, 
accelerated by the hot air below ; the fire blew up the hydrogen 
and the machine was shattered. A commemoration obelisk is to 
be seen at Wimille, near Boulogne. 

In balloon calculations it is convenient to suppose the total 
weight, W lb, to be distributed over the skin of the envelope, at 
a superficial density m = W/S, lb/ft 2 , S denoting the surface 
in ft 2 . 

Then, neglecting W as insensible, 

VA — mS is the lift at the ground, 
VA £ - mS, 

at a height z, where the density has fallen from p to p z . 
The balloon will rise to this height z, if 

(14) VA P± = mS, VA = qmS, q = £-, 

f> Pz 

M.F. T 



114 THE DYNAMICS OF MECHANICAL FLIGHT 

For a spherical balloon of diameter d ft, 

(15) V = A^3, s = ttcP, d = 6 ^, 

6 A 

6 T 'Af v - w „ 2 3 ^ 3 



(16) VA = qnw (— )*, 



7 = 36 



— 



A*' 



called by the French the equation of the " three cubes " ; but it is 
simpler to work to the diameter d in (15). 

The calculation is given on p. 337, Hydrostatics, required by 
the Jesuit Francis Lana, 1670, for his idea of a copper cylinder, 
exhausted of air, to serve as a balloon ; the exhaustion can be 
carried out by boiling a small quantity of air inside until the 
steam has driven out all the air ; then when sealed up a vacuum 
would be formed when the steam was condensed. 

Thus with copper, 0*01 inch thick, we can take m = J lb/ft 2 , 
and with A = 0*08 lb/ft 3 , d = 37'5q; so that on the ground, 
with q = 1, d = 37 '5 ft, and the copper sphere is just about to 
rise ; and taking d = 100 ft, 

q 6m 3' P 8' 
about 5 miles high in an isothermal atmosphere. 

The idea was very creditable, but it did not reckon with the 
collapsing pressure of the atmosphere ; and at that date, 1670, 
hydrogen was unknown, and even the name gas had not been 
invented ; and it was only the accidental observation made by 
Montgolfier mentioned above which showed, 100 years later, 
that the balloon problem was feasible, with a light envelope, 
where the gauge pressure in the interior was very low and the 
stress in the envelope correspondingly small. 

The gauge pressure (excess over atmospheric pressure outside) 
at a height y ft above the lower end of the neck or appendix tube 
will be 

(17) (p - £) * = Ay, lb/W, 



PNEUMATICAL PRINCIPLES OF AN AIR SHIP 115 



at the ground, changing to — Ay at a height z in the air ; this 
is neglecting the variation of density of the air and gas in a 



height so small comparatively as y ft. 



For the cylindrical gas bag on an air ship of diameter d ft, and 
p diameters long (Fig. 57), we may put 



(18) 
(19) 

(20) 



V = I 7r<7 s jj, S = - d% 
J wtPpA = qm' 3 S = qm-d-p, d ■■ 

r=i6^^ 3 . 



4 mq 




Thus for an air ship 500 ft long by 50 ft diameter, d 
p = 10, weighing 25 tons, 



= 50. 



(21) 



25 x 2,210 



- x 500 x 50 
and rilled with hydrogen of n = 14. 
(22) ^(l-^W =|L = 0-075 Ib/fta, 



(23) 



(24) 



= ^L = ^L v 5Q x ^ x 5QQ x 50 = 1>3 
q ~ ±m ~ 700 > 4 x 25 x 2,240 



2-3 fclogiotf =7,000 ft, 



the height ascended in an isothermal atmosphere, 
k = 27,000 ft. 



with 



116 THE DYNAMICS OF MECHANICAL FLIGHT 

In the non-rigid air ship the cylindrical envelope forms one 
continuous gas hag. and this must he kept constantly inflated. 

If this should become partially deflated, like an ordinary 
spherical balloon, the indentation would affect the stability as it 
runs from one end of the belly to the other ; and in the middle 
it would cause the gas bag to lose longitudinal stiffness, and 
allow it to double up like a bolster, as in Fig. 58. 




Fig-. 58. 

A ballonet is required, and this is an interior bag which can 
be inflated by an air-pump to fill up the space left vacant by 
the escape of the hydrogen through the appendix in the ascent, 
and to keep the balloon distended during the descent, and on 
the ground, in spite of gas leakage. 

One ballonet at each end inside is useful, shown by the dotted 
lines in Fig. 57, to serve like a trimming tank of a steamer, for 
keeping the air ship on an even keel. 



In the air ship of rigid type the hydrogen is carried in a 
number of separate gas bags enclosed in a lattice frame cage, 
and covered with a smooth envelope to diminish frictional drag, 
so that this type is equivalent to a series of spherical balloons 
harnessed together in a horizontal line. 

The gondolas are suspended below, as in Fig. 57, carrying the 
engines and crew. 

Consider the power required to drive this air ship at a given 
speed, Q f/s, or S rub* against the resistance of skin friction 
of the air. 



PNEUMATIC AL PRINCIPLES OF AN AIR SHIP 117 

It is estimated as the result of experiment that frictional drag 
is' about— ^ of pressure due to normal impact, taken above at 



|) 2 ib/ ft - 



so the friction is taken at 



400 



lb ft 2 . 



Caution is required here not to take double or half this value ; 
as experimenters have an awkward habit of tabulating frictional 
drag sometimes for both sides of the surface in their experiments. 

Here S = it X 50 X 500 = 78.540. say 80,000 ft 2 : 



(25) 

(26) 



F 



s m 



T.H.P. 



i* 



lb. 



FQ j^ 

330 660" 



Thus, if a speed of 45 m h is required 
Q = 66 



T . H .p. = |g _ 458 . 6 . 



With screws working at 25 per cent. slip, their efficiency is 

0'75, 75 %, so that S.H.P. = 575, requiring probably 700 I.H.P. 
or more ; or in a tabular form. 



s 


30 


Q 


44 


S.H.P. 


170 



36 
53 
390 



40 
58 
390 



45 
66 
575 



so that 860 I. H. P. should be expected to give a speed somewhere 
between 30 and 36 m/h. 



FINIS. 



INDEX 



PAGE 
2 
8 
1 
88 
104 
115 
115 
2 
81 
67 
67 
86 
114 
1 
25 



Abaeis 

Aeroplane 

vEschylus ... 

Alteration of trim ... 

Air propeller 

Air ship 109 

Air ship , non-rigid 

Alexander the Great 

Angular velocity ... 
Anomaly, mean 
Anomaly, true 
Apparent gravity ... 

Appendix tube 

Archytas of Tarentum 
Aspect ratio... 
Atmospheric pressure in past 
Geological Ages 4 

Ballonet 116 

Basset 42 

Bleriot 1, 82 

Borda mouthpiece ... ... ... 59 

Boyle 108 

Bronze horse 2 

Camber 105 

Cambered wing ... ... ... 40 

Cavitation 97 

Centre of pressure ... ... 23,40 

Centrifugal force ... ... ... 98 

Centrifugal pull 99 

Centrifugal pump ... ... ... 104 

Cervantes ... ... ... ... 2 

Charles 108 

Charlo-Montgolfiere 113 

Chateau de la Muette ... ... 107 

Chaucer ... ... ... ... 2 

Chavez 58 



Circular function ... 


... 65 


Clifford 


... 6, 10 


Cody 


... 82 


Conformal representation (map- 


Piog) ••• 


... 14 


Conjugate hyperbola 


... 68 


Continuity, equation of ... 


... 101 


Contraction, coefficient of 


... 62 


Coriolis 


... 97 


Cosmos 


4 


Counter current 


... 22 


Cyrano de Bergerac 


4 


DAEDALUS 


1 


Damped vibration ... 


... 72 


Demeter 


1 


Demoiselle (dragon-fly) ... 


5 


Denkschrift der I. L. A. ... 


1 


Disc area 


... 90 


Drag, frictional 


... 116 


Dynamical lemma ... 


... 74 


Dynamic head 


38, 40 


Dynamics, linear ... 


... 86 


Dynamics, notes on 


... 85 


Eddy whirlwind ... 


... 58 


Elastica 


... 58 


Electrical applications 


... 32 


Electrical law of flow 


... 41 


Ellipse 


... 68 


Elliptic cylinder ... 


... 42 


Elliptic integral 


... 56 


Empirical factor 


... 90 


Equation of continuity ... 


... 101 


Equation of the "three cubes ' 


... 114 


Equivalent pendulum 


75, 79 


Estimate of petrol ... 


... 27 



1-20 



INDEX 





PAGE 




PAGE 


Faeiian 


... 82 


Integral calculus ... 


.. 64 


Finite breadth, stream of.. 


... 48 


Integral, general irrational 


.. 67 


Fitzgerald ... 


I 


Integrals, typical irrational 


.. 67 


Flight 


14 


Integration... 


.. 72 


Forth Bridge 


... 17 


Integration round a conic 


.. 67 


Francis Lana 


... 114 






Frictional drag 


... 116 






Function, circular... 


... 65 


Kelvin 


.. 80 


Function, hyperbolic 


... 65 


Kinetic theory of a gas . . . 


.. 96 






Kirchhoff 


14, 19 


Gas equation 


... 108 






Gauge pressure 


38, 114 


Lag, lead and 


.. 72 


Gnome motor 


82, 105 


Lamb 


.. 42 


Gondolas 


... 116 


Langley 


6 


Gradient, pressure ... 


... 95 


Langley's experiments 


.. 18 


Gravitation measure of force ... 8.7 


Law of the spring ... 


.. 88 


Gravity, apparent ... 


... 86 


Lead and lag 


.. 72 


Gyroscopic action 


... 82 


Left-handed screw 


.. 82 


Gyroscopic experiments .. 


... 73 


Lemma, dynamical 


... 74 






Lilienthal 


2 






Linear dynamics ... 


... 86 


Happy Valley 


3 


Locomotive 


... 11 


Harle 


4 


Love, A. E. H 


... 32 


Hecate 


2 






Helical strand of a rope . . 


... 94 






Helix 


... 93 


Macbeth 


2 


Helmholtz 


14, 19 


Marchis 


... 64 


Herbert Spencer 


... 6, 10 


Marquis d' Arlandes 


... 107 


Herodotus ... 


7 


" Matter and Motion," Maxwell 


s 74 


Hippisley ... 


... 41 


Mauretania ... 


... 104 


Homer 


1 


Maxim 


4, 6, 10 


Homogeneous atmosphere, 


height 


Maxwell's ' ' Matter and Motion 


74 


of the 


... 112 


Maxwell top 


73 


Horse power, shaft 


... 100 


Mich ell, J. T 


... 32 


Hot air balloon 


... 107 


Milton 


5 


Hyperbola ... 
Hyperbolic function 


... 68 


Moment of inertia ■•■ 


... 74 


... 65 


Momentum, lineal or angular 


74 


Hydraulic machine 


... 89 


Montgolfier ... 


107.114 


Hydrogen ... 


... 110 


Motion, second law of 


... 86 


Hydrostatics 


... 107 










Nappe dorsale 


22 


ICAEUS 


1 


Newton 


...4,14 


Impulse couple 


... 43 


Newton's second law of motion 


74 


Inertia, moment of 


... 74 


Non-rigid air ship ... 


... 115 


Injector flow 


... 63 


Normal incidence ... 


22 



INDEX 



121 



" Xotes on Dynamics " ... 
Xutation, precession and ... 



PAGE 

... 85 
... 84 



OsciLLATiox of the carriage body 



Paddle steamer ... 
Pelton wheel 
Pendulum ... 
Pendulum, equivalent 
Pendulum, simple equivalent 
Pier, wedge-shaped 
Pilatre de Rozier ... 

Pilcher 

Pitching, rolling and 

Planck 

Poinsot 

Pope... 

Precession 

Precession and nutation 
Preponderance 
Pressure, gauge 
Pressure gradient . . . 

Preussen 

Prolate spheroid 
Propeller, air 
Pseudo-Callisthenes 
Pump, centrifugal ... 
Pump, turbine 



Racing of the screw 

Rankine 

Rasselas 
Rayleigh 

Report 19 

Right-handed screw 
Roger Bacon 
Rolling and pitching 

Rota 

Rudder boxed in ... 



Sails of a windmill or ship 
Santos Dnmont 
M.F. 



83, 



107. 



103 
15 
74 

, 79 

75 
61 
113 
2 
83 
33 
84 



84 
74 

114 
95 
91 
45 

101 
2 

104 

102 



... nit; 

15. 91, 97 
2 

... 40 

19, 28 

77 

6 

... 83 

... 10.1 

50 



90 

82 



Schwarz-Christoffel 

Screw, racing of the 

Screw surface 

Second law of motion 

Shaft horse power . . . 

Slip or slip ratio 

Slip velocity 

Sphere, effective inertia of 

Spheroid, prolate ... 

Spring, law of the ... 

Stability of an elongated shot 

Static pressure head 

Steady motion 

Stream function 

Stream issuing from a channel 

Stream line past a plane barrier 



Tait. Thomson and 

Tank, trimming 

Tennyson ... 

Tension length 

Thomson and Tait ... 

Thomson's Researches, J. J 

" Three cubes," equation of the 

Tip velocity... 

Traction 

Trim, alteration of 
Trimming tank 
Turbine 
Turbine pump 



Vaxe of a weathercock 
Vector representation 
Telocity, angular ... 
Velocity function ... 
Velocity, slip 
Velocitv, tip 



YYellmax air ship 
Whirlwind, eddy . 
Windmill 
Wright 

Y\'ri2rht Brothers . 



PAGE 
14, 28 
106 
91 
86 
100 
94 
94 
45 
45 
88 
45 
3S 
78 
20 
59 
28 



42 

116 

5 

99 

42 

32 

114 
98 
63 
88 

116 
15 

102 



25 

74 
84 
'zQ 
94 
98 



113 
58 

L00 

82 

2 



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6 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 



*I 


00 





50 


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50 


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75 





50 





50 



Browne, R. E. Water Meters. (Science Series No. 8i.) i6mo, 

Bruce, E. M. Pure Food Tests i2mo, 

Bruhns, Dr. New Manual of Logarithms 8vo, half morocco, 

Brunner, R. Manufacture of Lubricants, Shoe Polishes and Leather 

Dressings. Trans, by C. Salter 8vo, 

Buel, R. H. Safety Valves. (Science Series No. 21.) i6mo, 

Bulman, H. F., and Redmayne, R. S. A. Colliery Working and Manage- 
ment 8vo, 

Burgh, N. P. Modern Marine Engineering 4to, half morocco, 

Burstall, F. W. Energy Diagram for Gas. With Text 8vo, 

Diagram. Sold separately *i 

Burt, W. A. Key to the Solar Compass i6mo, leather, 

Burton, F. G. Engineering Estimates and Cost Accounts i2mo, 

Buskett, E. W. Fire Assaying i2mo, 

Byers, H. G., and Knight, H. G. Notes on Qualitative Analysis . . . .8vo, 

Cain, W. Brief Course in the Calculus i2mo, 

Elastic Arches. (Science Series No. 48.) i6mo, 

Maximum Stresses. (Science Series No. 38.) i6mo, 

Practical Designing Retaining of Walls. (Science Series No. 3.) 

i6mo, o 50 
Theory of Steel-concrete Arches and of Vaulted Structures. 

(Science Series No. 42.) i6mo, 

Theory of Voussoir Arches. (Science Series No. 12.) i6mo, 

Symbolic Algebra. (Science Series No. 73.) i6mo, 

Campin, F. The Construction of Iron Roofs 8vo, 

Carpenter, F. D. Geographical Surveying. (Science Series No. 37.) . i6mo, 
Carpenter, R. C, and Diederichs, H. Internal Combustion Engines. 8vo, 
Carter, E. T. Motive Power and Gearing for Electrical Machinery . . 8vo, 

Carter, H. A. Ramie (Rhea), China Grass i2mo, 

Carter, H. R. Modern Flax, Hemp, and Jute Spinning 8vo, 

Cathcart, W. L. Machine Design. Part I. Fastenings 8vo, 

Cathcart, W. L., and Chaffee, J. I. Elements of Graphic Statics 8vo, 

Short Course in Graphics i2mo, 

Caven, R. M., and Lander, G. D. Systematic Inorganic Chemistry. i2mo, 

Chalkley, A. P. Diesel Engines 8vo, 

Chambers' Mathematical Tables 8vo, 

Charnock, G. F. Workshop Practice. (Westminster Series.). . . .8vo (In Press.) 

Charpentier, P. Timber. 8vo, *6 00 

Chatley, H. Principles and Designs of Aeroplanes. (Science Series.) 

No. 126.) i6mo, 

How to Use Water Power nmo, 

Child, C. D. Electric Arc 8vo, *(In 

Child, C. T. The How and Why of Electricity i2mo, 

Christie, W. W. Boiler- waters, Scale, Corrosion, Foaming 8vo, 

Chimney Design and Theory 8vo, 

Furnace Draft. (Science Series No. 123.) i6mo, 

Water: Its Purification for Use in the Industries 8vo, (In Press.) 

Church's Laboratory Guide. Rewritten by Edward Kinch 8vo, *2 50 

Clapperton, G. Practical Papermaking 8vo, 2 50 






50 





50 





50 


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50 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 7 

Clark, A. G. Motor Car Engineering. 

Vol. I. Construction *3 oo 

Vol. II. Design (In Press.) 

Clark, C. H. Marine Gas Engines nmo, *i 50 

Clark, D. K. Rules, Tables and Data for Mechanical Engineers 8vo, 5 00 

Fuel: Its Combustion and Economy nmo, 1 50 

The Mechanical Engineer's Pocketbook i6mo, 2 00 

Tramways: Their Construction and Working 8vo, 5 00 

Clark, J. M. New System of Laying Out Railway Turnouts i2mo, 1 00 

Clausen-Thue, W. ABC Telegraphic Code. Fourth Edition nmo, *5 00 

Fifth Edition 8vo, *7 00 

The A 1 Telegraphic Code 8vo, *7 50 

Cleemann, T. M. The Railroad Engineer's Practice i2mo, *i 50 

Clerk, D., and Idell, F. E. Theory of the Gas Engine. (Science Series 

No. 62.) i6mo, o 50 

Clevenger, S. R. Treatise on the Method of Government Surveying. 

i6mo, morocco 2 ^50 

Clouth, F. Rubber, Gutta-Percha, and Balata 8vo, *5 00 

Cochran, J. Treatise on Cement Specifications 8vo, (In Press.) . . 

Coffin, J. H. C. Navigation and Nautical Astronomy nmo, *3 50 

Colburn, Z., and Thurston, R. H. Steam Boiler Explosions. (Science 

Series No. 2.) i6mo, 

Cole, R. S. Treatise on Photographic Optics . i2mo, 

Coles-Finch, W. Water, Its Origin and Use 8vo, 

Collins, J. E. Useful Alloys and Memoranda for Goldsmiths, Jewelers. 

i6mo 

Constantine, E. Marine Engineers, Their Qualifications and Duties. 8vo, 

Coombs, H. A. Gear Teeth. (Science Series No. 120.) i6mo, 

Cooper, W. R. Primary Batteries 8vo, 

" The Electrician " Primers . . . • 8vo, 

Part I 

Part II 

Part III 

Copperthwaite, W. C. Tunnel Shields 4to, 

Corey, H. T. Water Supply Engineering 8vo (In Press.) 

Corfield, W. H. Dwelling Houses. (Science Series No. 50.) i6mo, 

Water and Water-Supply. (Science Series No. 17.) i6mo, 

Cornwall, H. B. Manual of Blow-pipe Analysis 8vo, 

Courtney, C. F. Masonry Dams 8vo, 

Cowell, W. B. Pure Air, Ozone, and Water nmo, 

Craig, T. Motion of a Solid in a Fuel. (Science Series No. 49.) .... i6mo, 

Wave and Vortex Motion. (Science Series No. 43.) i6mo, 

Cramp, W. Continuous Current Machine Design 8vo, 

Crocker, F. B. Electric Lighting. Two Volumes. 8vo. 

Vol. I. The Generating Plant 3 

Vol. II. Distributing Systems and Lamps 3 

Crocker, F. B., and Arendt, M. Electric Motors 8vo, 

Crocker, F. B., and Wheeler, S. S. The Management of Electrical Ma- 
chinery i2mo, *i 00 

Cross, C. F., Bevan, E. J., and Sindall, R. W. Wood Pulp and Its Applica- 
tions. (Westminster Series.) 8vo, *2 00 






50 


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50 


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50 


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50 



8 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Crosskey, L. R. Elementary Perspective 8vo, i oo 

Crosskey, L. R., and Thaw, J. Advanced Perspective 8vo, i 50 

Culley, J. L. Theory of Arches. (Science Series No. 87.) i6mo, o 50 

Davenport, C. The Book. (Westminster Series.) 8vo, *2 00 

Davies, D. C. Metalliferous Minerals and Mining 8vo, 5 00 

Earthy Minerals and Mining 8vo, 5 00 

Davies, E. H. Machinery for Metalliferous Mines 8vo, 8 00 

Davies, F. H. Electric Power and Traction 8vo, *2 00 

Dawson, P. Electric Traction on Railways 8vo, *o, 00 

Day, C. The Indicator and Its Diagrams i2mo, *2 00 

Deerr, N. Sugar and the Sugar Cane 8vo, *8 00 

Deite, C. Manual of Soapmaking. Trans, by S. T. King 4to, *5 00 

De la Coux, H. The Industrial Uses of Water. Trans, by A. Morris. 

8vo, 

Del Mar, W. A. Electric Power Conductors 8vo, 

Denny, G. A. Deep-level Mines of the Rand 4to, 

Diamond Drilling for Gold 

De Roos, J. D. C. Linkages. (Science Series No. 47.) i6mo, 

Derr, W. L. Block Signal Operation Oblong nmo, 

Maintenance-of-Way Engineering (In Preparation.) 

Desaint, A. Three Hundred Shades and How to Mix Them 8vo, 

De Varona, A. Sewer Gases. (Science Series No. 55.) i6mo, 

Devey, R. G. Mill and Factory Wiring. (Installation Manuals Series.) 

i2mo, 

Dibdin, W. J. Public Lighting by Gas and Electricity 8vo, 

Purification of Sewage and Water ■ 8vo, 

Dichmann, Carl. Basic Open-Hearth Steel Process i2mo, 

Dieterich, K. Analysis of Resins, Balsams, and Gum Resins 8vo, 

Dinger, Lieut. H. C. Care and Operation of Naval Machinery nmo, 

Dixon, D. B. Machinist's and Steam Engineer's Practical Calculator. 

i6mo, morocco, 1 25 
Doble, W. A. Power Plant Construction on the Pacific Coast (In Press.) 
Dodd, G. Dictionary of Manufactures, Mining, Machinery, and the 

Industrial Arts i2mo, 1 50 

Dorr, B. F. The Surveyor's Guide and Pocket Table-book. 

i6mo, morocco, 2 00 

Down, P. B. Handy Copper Wire Table i6mo, *i 00 

Draper, C. H. Elementary Text-book of Light, Heat and Sound. . . i2mo, 1 00 

Heat and the Principles of Thermo-dynamics i2mo, 1 50 

Duckwall, E. W. Canning and Preserving of Food Products 8vo, *5 00 

Dumesny, P., and Noyer, J. Wood Products, Distillates, and Extracts. 

8vo, *4 50 
Duncan, W. G., and Penman, D. The Electrical Equipment of Collieries. 

8vo, 
Dunstan, T A. E., and Thole, F. B. T. Textbook of Practical Chemistry. 

i2mo, 
Duthie, A. L. Decorative Glass Processes. (Westminster Series.). .8vo, 

Dwight, H. B. Transmission Line Formulas 8vo, (In 

Dyson, S. S. Practical Testing of Raw Materials 8vo, 

Dyson, S. S., and Clarkson, S. S. Chemical Works 8vo, 



*4 


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50 



D. VAX X03TRAXD COMPANY'S SHORT TITLE CATALOG 9 

Eccles, R. G., and Duckwall, E. W. Food Preservatives 8vo, paper o 50 

Eddy, H. T. Researches in Graphical Statics 8vo, 1 50 

Maximum Stresses under Concentrated Loads 8vo, 1 50 

Edgcumbe, K. Industrial Electrical Measuring Instruments 8vo, *2 50 

Eissler, M. The Metallurgy of Gold 8vo 7 50 

The Hydrometallurgy of Copper 8vo, *4 50 

The Metallurgy of Silver 8vo, 4 00 

The Metallurgy of Argentiferous Lead 8vo, 5 00 

Cyanide Process for the Extraction of Gold 8vo, 3 00 

A Handbook on Modern Explosives 8vo, 5 00 

Ekin, T. C. Water Pipe and Sewage Discharge Diagrams folio, *3 00 

Eliot, C. W., and Storer, F. H. Compendious Manual of Qualitative 

Chemical Analysis nmo, *i 25 

Elliot, Major G. H. European Light-house Systems 8vo, 5 00 

Ennis, Wm. D. Linseed Oil and Other Seed Oils 8vo, *4 00 

Applied Thermodynamics 8vo *4 50 

Flying Machines To-day nmo, *i 50 

Vapors for Heat Engines i2mo, *i 00 

Erfurt, J. Dyeing of Paper Pulp. Trans, by J. Hubner 8vo, *7 50 

Erskine-Murray, J. A Handbook of Wireless Telegraphy 8vo, *3 50 

Evans, C. A. Macadamized Roads (In Press.) 

Ewing, A. J. Magnetic Induction in Iron 8vo, *4 00 

Fairie, J. Notes on Lead Ores nmo, *i 00 

Notes on Pottery Clays nmo, *i 50 

Fairley, W., and Andre, Geo. J. Ventilation of Coal Mines. (Science 

Series No. 58.) i6mo, o 50 

Fairweather, W. C. Foreign and Colonial Patent Laws 8vo, *3 00 

Fanning, J. T. Hydraulic and Water-supply Engineering 8vo, *5 00 

Fauth, P. The Moon in Modern Astronomy. Trans, by J. McCabe. 

8vo, *2 00 

Fay, I. W. The Coal-tar Colors 8vo, *4 00 

Fernbach, R. L. Glue and Gelatine 8vo, *3 00 

Chemical Aspects of Silk Manufacture i2mo, *i 00 

Fischer, E. The Preparation of Organic Compounds. Trans, by R. V. 

Stanford nmo, *i 25 

Fish, J. C. L. Lettering of Working Drawings Oblong 8vo, 1 00 

Fisher, H. K. C, and Darby, W. C. Submarine Cable Testing 8vo, *3 50 

Fiske, Lieut. B. A. Electricity in Theory and Practice 8vo, 2 50 

Fleischmann, W. The Book of the Dairy. Trans, by C. M. Aikman. 8vo, 4 00 
Fleming, J. A. The Alternate-current Transformer. Two Volumes. 8vo. 

Vol. I. The Induction of Electric Currents *5 00 

Vol. II. The Utilization of Induced Currents *5 00 

Propagation of Electric Currents 8vo, *3 00 

Centenary of the Electrical Current 8vo, *o 50 

Electric Lamps and Electric Lighting 8vo, *3 00 

Electrical Laboratory Notes and Forms 4to, *5 00 

A Handbook for the Electrical Laboratory and Testing Room. Two 

Volumes 8vo, each, *5 00 

Fluery, H. The Calculus Without Limits or Infinitesimals. Trans, by 
C. 0. Mailloux (In Press.) 



1U D. VAN NOSTKAND COMPANY'S SHORT TITLE CATALOG 

nynn, P. J. Flow of Water. (Science Series No. 84.) i6mo, o 50 

Hydraulic Tables. (Science Series No. 66.) i6mo, 50 

Foley, N, British and American Customary and Metric Measures .. folio, *3 00 
Foster, H. A. Electrical Engineers' Pocket-book. (Sixth Edition.) 

i2mo, leather, 5 00 

Engineering Valuation of Public Utilities and Factories 8vo, *3 00 

Foster, Gen. J. G. Submarine Blasting in Boston (Mass.) Harbor.. . 4to, 3 50 

Fowle, F. F. Overhead Transmission Line Crossings i2mo, *i 50 

The Solution of Alternating Current Problems 8vo (In Press.) 

Fox, W. G. Transition Curves. (Science Series No. no.) i6mo, o 50 

Fox, W., and Thomas, C. W. Practical Course in Mechanical Draw- 
ing i2mo, 1 25 

Foye, J. C. Chemical Problems. (Science Series No. 69.) i6mo, o 50 

Handbook of Mineralogy. (Science Series No. 86.) i6mo, o 50 

Francis, J. B. Lowell Hydraulic Experiments 4to, 15 00 

Freudemacher, P. W. Electrical Mining Installations. (Installation 

Manuals Series ) . i2mo, *i 00 

Frith, J. Alternating Current Design 8vo, *2 00 

Fritsch, J. Manufacture of Chemical Manures. Trans, by D. Grant. 

8vo, *4 00 

Frye, A. I. Civil Engineers' Pocket-book i2mo, leather, 

Fuller, G. W. Investigations into the Purification of the Ohio River. 

4to. *io 00 

Furnell, J. Paints, Colors, Oils, and Varnishes 8vo, *i 00 

Gairdner, J. W. I. Earthwork 8vo, (In Press.) 

Gant, L. W. Elements of Electric Traction 8vo, *2 50 

Garforth, W. E. Rules for Recovering Coal Mines after Explosions and 

Fires. nmo, leather, 1 50 

Gaudard, J. Foundations. (Science Series No. 34.) i6mo, o 50 

Gear, H. B., and Williams, P. F. Electric Central Station Distribution 

Systems 8vo, *3 00 

Geerligs, H. C. P. Cane Sugar and Its Manufacture 8vo, *5 00 

Geikie, J. Structural and Field Geology 8vo, *4 00 

Gerber, N. Analysis of Milk, Condensed Milk, and Infants' Milk-Food. 8vo, 1 25 
Gerhard, W. P. Sanitation, Watersupply and Sewage Disposal of Country 

Houses i2mo, *2 00 

Gas Lighting. (Science Series No. in.) i6mo, o 50 

Household Wastes. (Science Series No. 97.) i6mo, o 50 

House Drainage. (Science Series No. 63.) i6mo, 50 

Sanitary Drainage of Buildings. (Science Series No. 93.) .... i6mo, 50 

Gerhardi, C. W. H. Electricity Meters 8vo, *4 00 

Geschwind, L. Manufacture of Alum and Sulphates. Trans, by C. 

Salter 8vo, *s 00 

Gibbs, W. E. Lighting by Acetylene nmo, *i 50 

Physics of Solids and Fluids. (Carnegie Technical School's Text- 
books.) *i 50 

Gibson, A. H. Hydraulics and Its Application 8vo, *5 00 

Water Hammer in Hydraulic Pipe Lines i2mo, *2 00 

Gilbreth, F. B. Motion Study nmo, *2 00 

Primer of Scientific Management nmo, *i 00 



4 


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50 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 11 

Gillmore, Gen. Q. A. Limes, Hydraulic Cements ar d Mortars 8vo, 

Roads, Streets, and Pavements i2mo, 

Golding, H. A. The Theta-Phi Diagram i2mo, 

Goldschmidt, R. Alternating Current Commutator Motor 8vo, 

Goodchild, W. Precious Stones. (Westminster Series.) 8vo, 

Goodeve, T. M. Textbook on the Steam-engine nmo, 

Gore, G. Electrolytic Separation of Metals 8vo, 

Gould, E. S. Arithmetic of the Steam-engine nmo, 

Calculus. (Science Series No. 112.) i6mo, 

High Masonry Dams. (Science Series No. 22.) i6mo, 

Practical Hydrostatics and Hydrostatic Formulas. (Science Series 

No. 117.) i6mo, o 50 

Grant, J. Brewing and Distilling. (Westminster Series.) 8vo (In Press.) 

Gratacap, L. P. A Popular Guide to Minerals 8vo (In Press.) 

Gray, J. Electrical Influence Machines i2mo, 2 00 

Marine Boiler Design i2mo, (In Press.) 

Greenhill, G. Dynamics of Mechanical Flight 8vo, (In Press.) 

Greenwood, E. Classified Guide to Technical and Commercial Books. 8vo, *3 00 

Gregorius, R. Mineral Waxes. Trans, by C. Salter i2mo, *3 00 

Griffiths, A. B. A Treatise on Manures nmo, 3 00 

Dental Metallurgy 8vo, *3 50 

Gross, E. Hops 8vo, *4 50 

Grossman, J. Ammonia and Its Compounds nmo, *i 25 

Groth, L. A. Welding and Cutting Metals by Gases or Electricity. . . .8vo, *3 00 

Grover, F. Modern Gas and Oil Engines 8vo, *2 00 

Gruner, A. Power-loom Weaving 8vo, *3 00 

Giildner, Hugo. Internal Combustion Engines. Trans, by H. Diederichs. 

4to, *io 00 

Gunther, C. 0. Integration nmo, *i 25 

Gurden, R. L. Traverse Tables folio, half morocco, *7 50 

Guy, A. E. Experiments on the Flexure of Beams 8vo, *i 25 

Haeder, H. Handbook on the Steam-engine. Trans, by H. H. P. 

Powles •. nmo, 3 00 

Hainbach, R. Pottery Decoration. Trans, by C. Slater nmo, *3 00 

Haenig, A. Emery and Emery Industry 8vo, (In Press.) 

Hale, W. J. Calculations of General Chemistry nmo, *i 00 

Hall, C. H. Chemistry of Paints and Paint Vehicles nmo, *2 00 

Hall, R. H. Governors and Governing Mechanism nmo, *2 00 

Hall, W. S. Elements of the Differential and Integral Calculus 8vo, *2 25 

Descriptive Geometry 8vo volume and a 4to atlas, *3 50 

Haller, G. F., and Cunningham, E. T. The Tesla Coil nmo, *i 25 

Halsey, F. A. Slide Valve Gears nmo, 1 50 

The Use of the Slide Rule. (Science Series No. 114.) i6mo, o 50 

Worm and Spiral Gearing. (Science Series No. 116.) i6mo, o 50 

Hamilton, W. G. Useful Information for Railway Men i6mo, 1 00 

Hammer, W. J. Radium and Other Radio-active Substances 8vo, *i 00 

Hancock, H. Textbook of Mechanics and Hydrostatics 8vo, 1 50 

Hardy, E. Elementary Principles of Graphic Statics nmo, *i 50 

Harrison, W. B. The Mechanics' Tool-book nmo, 1 50 

Hart, J. W. External Plumbing Work 8vo, *3 00 



12 D. VAN.NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Hart, J. W. Hints to Plumbers on Joint Wiping 8vo, *3 oo 

Principles of Hot Water Supply 8vo, *3 oo 

Sanitary Plumbing and Drainage 8vo, *3 oo 

Haskins, C. H. The Galvanometer and Its Uses i6mo, i 50 

Hatt, J. A. H. The Colorist square nmo, *i 50 

Hausbrand, E. Drying by Means of Air and Steam. Trans, by A. C. 

Wright i2mo, *2 00 

Evaporating, Condensing and Cooling Apparatus. Trans, by A. C. 

Wright 8vo, *5 00 

Hausner, A. Manufacture of Preserved Foods and Sweetmeats. Trans. 

by A. Morris and H. Robson 8vo, 

Hawke, W. H. Premier Cipher Telegraphic Code 4to, 

100,000 Words Supplement to the Premier Code 4to, 

Hawkesworth, J. Graphical Handbook for Reinforced Concrete Design. 

4to, 

Hay, A. Alternating Currents 8vo, 

Electrical Distributing Networks and Distributing Lines 8vo, 

Continuous Current Engineering 8vo, 

Heap, Major D. P. Electrical Appliances 8vo, 

Heaviside, 0. Electromagnetic Theory. Two Volumes 8vo, each, 

Heck, R. C. H. The Steam Engine and Turbine 8vo, 

Steam-Engine and Other Steam Motors. Two Volumes. 

Vol. I. Thermodynamics and the Mechanics 8vo, 

Vol. II. Form, Construction, and Working 8vo, 

Notes on Elementary Kinematics 8vo, boards, 

Graphics of Machine Forces 8vo, boards, 

Hedges, K. Modern Lightning Conductors 8vo, 

Heermann, P. Dyers' Materials. Trans, by A. C. Wright nmo, 

Hellot, Macquer and D'Apiigny. Art of Dyeing Wool, Silk and Cotton. 

8vo, 

Henrici, 0. Skeleton Structures 8vo, 

Hering, D. W. Essentials of Physics for College Students 8vo, 

Hering-Shaw, A. Domestic Sanitation and Plumbing. Two Vols.. .8vo, 

Elementary Science 8vo, 

Herrmann, G. The Graphical Statics of Mechanism. Trans, by A. P. 

Smith nmo, 

Herzfeld, J. Testing of Yarns and Textile Fabrics 8vo, 

Hildebrandt, A. Airships, Past and Present 8vo, 

Hildenbrand, B. W. Cable-Making. (Science Series No. 32.) i6mo, 

Hilditch, T. P. A Concise History of Chemistry i2mo, 

Hill, J. W. The Purification of Public Water Supplies. New Edition. 

(In Press.) 

Interpretation of Water Analysis (In Press.) 

Hiroi, I. Plate Girder Construction. (Science Series No. 95.) i6mo, 

Statically-Indeterminate Stresses nmo, 

Hirshfeld, C. F. Engineering Thermodynamics. (Science Series No. 45.) 

i6mo, 

Hobart, H. M. Heavy Electrical Engineering 8vo, 

Design of Static Transformers i2mo, 

Electricity 8vo, 

Electric Trains 8vo, 



"3 


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D. VAN XOSTRAXD COMPANY'S SHORT TITLE CATALOG 13 

Hobart, H. M. Electric Propulsion of Ships 8vo, *2 oo 

Hobart, J. F. Hard Soldering, Soft Soldering and Brazing, nmo, 

(In Press.) 

Hobbs, W. R. P. The Arithmetic of Electrical Measurements nmo, 

Hoff, J. N. Paint and Varnish Facts and Formulas i2mo, 

Hoff, Com. W. B. The Avoidance of Collisions at Sea. . . i6mo, morocco, 

Hole, W. The Distribution of Gas 8vo, 

Holley, A. L. Railway Practice folio, 

Holmes, A. B. The Electric Light Popularly Explained .... nmo, paper, 

Hopkins, N. M. Experimental Electrochemistry 8vo, 

Model Engines and Small Boats nmo, 

Hopkinson, J. Shoolbred, J. N., and Day, R. E. Dynamic Electricity. 

(Science Series No. 71.) i6mo, 

Horner, J. Engineers' Turning 8vo, 

Metal Turning nmo, 

Toothed Gearing nmo, 

Houghton, C. E. The Elements of Mechanics of Materials nmo, 

Houllevigue, L. The Evolution of the Sciences 8vo, 

Howe, G. Mathematics for the Practical Man nmo, 

Howorth, J. Repairing and Riveting Glass, China and Earthenware. 

8vo, paper, 

Hubbard, E. The Utilization of Wood- waste 8vo, 

Hiibner, J. Bleaching and Dyeing of Vegetable and Fibrous Materials 

(Outlines of Industrial Chemistry) 8vo, (In Press.) 

Hudson, O. F. Iron and Steel. (Outlines of Industrial Chemistry.) 

8vo, (In Press.) 

Humper, W. Calculation of Strains in Girders nmo, 

Humphreys, A. C. The Business Features of Engineering Practice . 8vo, 

Hunter, A. Bridge Work 8vo, {In Press.) 

Hurst, G. H. Handbook of the Theory of Color 8vo, 

Dictionary of Chemicals and Raw Products 8vo, 

Lubricating Oils, Fats and Greases 8vo, 

Soaps 8vo, 

Textile Soaps and Oils 8vo, 

Hurst, H. E., and Lattey, R. T. Text-book of Physics 8vo, 

Hutchinson, R. W., Jr. Long Distance Electric Power Transmission. 

nmo, *3 00 
Hutchinson, R. W., Jr., and Ihlseng, M. C. Electricity in Mining. . nmo, 

(In Press) 
Hutchinson, W. B. Patents and How to Make Money Out of Them, nmo, 

Hutton, W. S. Steam-boiler Construction 8vo, 

Practical Engineer's Handbook 8vo, 

The Works' Manager's Handbook 8vo, 

Hyde, E. W. Skew Arches. (Science Series No. 15.) i6mo, 

Induction Coils. (Science Series No. 53.) i6mo, 

Ingle, H. Manual of Agricultural Chemistry 8vo, 

Innes, C. H. Problems in Machine Design nmo, 

Air Compressors and Blowing Engines nmo, 

Centrifugal Pumps nmo, 

The Fan 1 2mo, 



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14 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Isherwood, B. F. Engineering Precedents for Steam Machinery 8vo, 

Ivatts, E. B. Railway Management at Stations 8vo, 

Jacob, A., and Gould, E. S. On the Designing and Construction of 

Storage Reservoirs. (Science Series No. 6.) i6mo, 

Jamieson, A. Text Book on Steam and Steam Engines 8vo, 

Elementary Manual on Steam and the Steam Engine i2mo, 

Jannettaz, E. Guide to the Determination of Rocks. Trans, by G. W. 

Plympton i2mo, 

Jehl, F. Manufacture of Carbons 8vo, 

Jennings, A. S. Commercial Paints and Painting. (Westminster Series.) 

8vo (In Press.) 

Jennison, F. H. The Manufacture of Lake Pigments 8vo, 

Jepson, G. Cams and the Principles of their Construction 8vo, 

Mechanical Drawing 8vo (In Preparation.) 

Jockin, W. Arithmetic of the Gold and Silversmith i2mo, 

Johnson, G. L. Photographic Optics and Color Photography 8vo, 

Johnson, J. H. Arc Lamps and Accessory Apparatus. (Installation 

Manuals Series.) i2mo, *o 75 

Johnson, T. M. Ship Wiring and Fitting. (Installation Manuals 

Series) i2mo, 

Johnson, W. H. The Cultivation and Preparation of Para Rubber. . . 8vo, 

Johnson, W. McA. The Metallurgy of Nickel (In Preparation.) 

Johnston, J. F. W., and Cameron, C. Elements of Agricultural Chemistry 

and Geology nmo, 

Joly, J. Raidoactivity and Geology i2mo, 

Jones, H. C. Electrical Nature of Matter and Radioactivity i2mo, 

Jones, M. W. Testing Raw Materials Used in Paint i2mo, 

Jones, L., and Scard, F. I. Manufacture of Cane Sugar 8vo, 

Jordan, L. C. Practical Railway Spiral nmo, Leather, (In Press.) 

Joynson, F. H. Designing and Construction of Machine Gearing. . . .8vo, 
Jiiptner, H. F. V. Siderology: The Science of Iron 8vo, 

Kansas City Bridge 4to, 

Kapp, G. Alternate Current Machinery. (Science Series No. 96.) . i6mo, 

Electric Transmission of Energy , i2mo, 

Keim, A. W. Prevention of Dampness in Buildings 8vo, 

Keller, S. S. Mathematics for Engineering Students, nmo, half leather. 

Algebra and Trigonometry, with a Chapter on Vectors *i 

Special Algebra Edition 

Plane and Solid Geometry 

Analytical Geometry and Calculus 

Kelsey, W. R. Continuous-current Dynamos and Motors 8vo, 

Kemble, W. T., and Under hill, C. R. The Periodic Law and the Hydrogen 

Spectrum 8vo, paper, 

Kemp, J. F. Handbook of Rocks 8vo, 

Kendall, E. Twelve Figure Cipher Code 4to, 

Kennedy, A. B. W., and Thurston, R. H. Kinematics of Machinery. 

(Science Series No. 54.) i6mo, 50 

Kennedy, A. B. W., Unwin, W. C, and Idell, F. E. Compressed Air. 

(Science Series No. 106.) i6mo, 50 



*0 


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D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 15 

Kennedy, R. Modern Engines and Power Generators. Six Volumes. 4to, 

Single Volumes each, 

Electrical Installations. Five Volumes 4to, 

Single Volumes each, 

Flying Machines; Practice and Design i2mo, 

Principles of Aeroplane Construction 8vo, 

Kennelly, A. E. Electro-dynamic Machinery 8vo, 

Kent, W. Strength of Materials. (Science Series No. 41.) i6mo, 

Kershaw, J. B. C. Fuel, Water and Gas Analysis 8vo, 

Electrometallurgy. (Westminster Series.) 8vo, 

The Electric Furnace in Iron and Steel Production i2mo, 

Kinzbrunner, C. Alternate Current Windings 8vo, 

Continuous Current Armatures 8vo, 

Testing of Alternating Current Machines ... 8vo, 

Kirkaldy, W. G. David Kirkaldy's System of Mechanical Testing 4to, 

Kirkbride, J. Engraving for Illustration : 8vo, 

Kirkwood, J. P. Filtration of River Waters 4to, 

Klein, J. F. Design of a High-speed Steam-engine 8vo, 

Physical Significance of Entropy 8vo, 

Kleinhans, F. B. Boiler Construction 8vo, 

Knight, R.-Adm. A. M. Modern Seamanship 8vo, 

Half morocco 

Knox, W. F. Logarithm Tables {In Preparation.) 

Knott, C. G., and Mackay, J. S. Practical Mathematics 8vo, 

Koester, F. Steam-Electric Power Plants . .4to, 

Hydroelectric Developments and Engineering 4to, 

Koller, T. The Utilization of Waste Products 8vo, 

Cosmetics 8vo, 

Kretchmar, K. Yarn and Warp Sizing 8vo, 

Krischke, A. Gas and Oil Engines i2mo, 

Lambert, T. Lead and its Compounds 8vo, 

Bone Products and Manures 8vo, 

Lamborn, L. L. Cottonseed Products 8vo, 

Modern Soaps, Candles, and Glycerin 8vo, 

Lamprecht, R. Recovery Work After Pit Fires. Trans, by C. Salter . . 8vo, 
Lanchester, F. W. Aerial Flight. Two Volumes. 8vo. 

Vol. I. Aerodynamics *6 

Aerial Flight. Vol. II. Aerodonetics 

Larner, E. T. Principles of Alternating Currents nmo, 

Larrabee, C. S. Cipher and Secret Letter and Telegraphic Code i6mo, 

La Rue, B. F. Swing Bridges. (Science Series No. 107.) i6mo, 

Lassar-Cohn, Dr. Modern Scientific Chemistry. Trans, by M. M. Patti- 

son Muir nmo, 

Latimer, L. H., Field, C. J., and Howell, J. W. Incandescent Electric 

Lighting. (Science Series No. 57.) i6mo, 

Latta, M. N. Handbook of American Gas-Engineering Practice 8vo, 

American Producer Gas Practice 4to, 

Leask, A. R. Breakdowns at Sea nmo, 

Refrigerating Machinery i2mo, 

Lecky, S. T. S. " Wrinkles " in Practical Navigation 8vo, 



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16 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Le Doux, M. Ice-Making Machines. (Science Series No. 46.) .... i6mo, o 5c 

Leeds, C. C. Mechanical Drawing foi Trade Schools oblong 4to, 

High School Edition *i 25 

Machinery Trades Edition *2 00 

Lefe'vre, L. Architectural Pottery. Trans, by H. K. Bird and W. M. 

Binns 4to, 

Lehner, S. Ink Manufacture. Trans, by A. Morris and H. Robson . . 8vo, 

Lemstrom, S. Electricity in Agriculture and Horticulture 8vo, 

Le Van, W. B. Steam-Engine Indicator. (Science Series No. 78.) . i6mo, 
Lewes, V. B. Liquid and Gaseous Fuels. (Westminster Series.). .. .8vo, 

Lewis, L. P. Railway Signal Engineering 8vo, 

Lieber, B. F. Lieber's Standard Telegraphic Code 8vo, 

Code. German Edition 8vo, 

Spanish Edition 8vo, 

French Edition 8vo, 

Terminal Index " 8vo, 

Lieber's Appendix folio, 

Handy Tables 4to, 

Bankers and Stockbrokers' Code and Merchants and Shippers' Blank 

Tables 8vo, 

100,000,000 Combination Code 8vo, 

Engineering Code 8vo, 

Livermore, V. P., and Williams, J. How to Become a Competent Motor- 
man i2mo, 

Livingstone, R. Design and Construction of Commutators 8vo, 

Lobben, P. Machinists' and Draftsmen's Handbook 8vo, 

Locke, A. G. and C. G. Manufacture of Sulphuric Acid 8vo, 

Lockwood, T. D. Electricity, Magnetism, and Electro-telegraph .... 8vo, 

Electrical Measurement and the Galvanometer i2mo, 

Lodge, 0. J. Elementary Mechanics nmo, 

Signalling Across Space without Wires 8vo, 

Loewenstein, L. C, and Crissey, C. P. Centrifugal Pumps 

Lord, R. T. Decorative and Fancy Fabrics 8vo, 

Loring, A. E. A Handbook of the Electromagnetic Telegraph i6mo, 

Handbook. (Science Series No. 39.) i6mo, 

Low, D. A. Applied Mechanics (Elementary) i6mo, 

Lubschez, B J. Perspective (In Press.) 

Lucke, C. E.* Gas Engine Design 8vo, *3 00 

Power Plants: Design, Efficiency, and Power Costs. 2 vols. (In Preparation.) 

Lunge, G. Coal-tar and Ammonia. Two Volumes 8vo, *i5 00 

Manufacture of Sulphuric Acid and Alkali. Four Volumes 8vo, 

Vol. I. Sulphuric Acid. In two parts *i5 00 

Vol. II. Salt Cake, Hydrochloric Acid and Leblanc Soda. In two parts *i5 00 

Vol. III. Ammonia Soda *io 00 

Vol. IV. Electrolytic Methods (In Press.) 

■ Technical Chemists' Handbook i2mo, leather, *3 50 

Technical Methods of Chemical Analysis. Trans, by C. A. Keane. 

in collaboration with the corps of specialists. 

Vol. I. In two parts 8vo, *i5 00 

Vol. II. In two parts 8vo, *i8 00 

Vol. IH (In Preparation.) 



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80 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 17 

Lupton, A., Parr, G. D. A., and Perkin, H. Electricity as Applied to 

Mining 8vo, *4 50 

Luquer, L. M. Minerals in Rock Sections 8vo, *i 50 

Macewen, H. A. Food Inspection , 8vo, *2 50 

Mackenzie, N. F. Notes on Irrigation Works 8vo, *2 50 

Mackie, J. How to Make a Woolen Mill Pay 8vo, *2 00 

Mackrow, C. Naval Architect's and Shipbuilder's Pocket-book. 

i6mo, leather, 5 00 

Maguire, Wm. R. Domestic Sanitary Drainage and Plumbing 8vo, 4 00 

Mallet, A. Compound Engines. Trans, by R. R. Buel. (Science Series 

No. 10.) i6mo, 

Mansfield, A. N. Electro-magnets. (Science Series No. 64.) i6mo, o 50 

Marks, E. C. R. Construction of Cranes and Lifting Machinery .... i2mo, *i 50 

Construction and Working of Pumps i2mo, *i 50 

Manufacture of Iron and Steel Tubes nmo, *2 00 

Mechanical Engineering Materials nmo, *i 00 

Marks, G. C. Hydraulic Power Engineering 8vo, 3 50 

Inventions, Patents and Designs nmo, *i 00 

Marlow, T. G. Drying Machinery and Practice 8vo, *5 00 

Marsh, C. F. Concise Treatise on Reinforced Concrete 8vo, *2 50 

Reinforced Concrete Compression Member Diagram. Mounted on 

Cloth Boards *i 50 

Marsh, C. F., and Dunn, W. Reinforced Concrete 4to, *5 00 

Marsh, C. F., and Dunn, W. Manual of Reinforced Concrete and Con- 
crete Block Construction i6mo, morocco, *2 50 

Marshall, W. J., and Sankey, H. R. Gas Engines. (Westminster Series.) 

8vo, *2 00 

Martin. G, Triumphs and Wonders of Modern Chemistry 8vo, *2 00 

Martin, N. Properties and Design of Reinforced Concrete. 

(In Press.) 
Massie, W. W., and Underhill, C. R. Wireless Telegraphy and Telephony. 

nmo, *i 00 
Matheson, D. Australian Saw-Miller's Log and Timber Ready Reckoner. 

nmo, leather, 1 50 

Mathot, R. E. Internal Combustion Engines 8vo, *6 00 

Maurice, W. Electric Blasting Apparatus and Explosives 8vo, *3 50 

Shot Firer's Guide 8vo, *i 50 

Maxwell, J. C. Matter and Motion. (Science Series No. 36.) i6mo, o 50 

Maxwell, W. H., and Brown, J. T. Encyclopedia of Municipal and Sani- 
tary Engineering 4to, *io 00 

Mayer, A. M. Lecture Notes on Physics 8vo, 2 00 

McCullough, R. S. Mechanical Theory of Heat 8vo, 3 50 

Mcintosh, J. G. Technology of Sugar 8vo, *4 50 

Industrial Alcohol 8vo, *3 00 

Manufacture of Varnishes and Kindred Industries. Three Volumes. 

8vo. 

Vol. I. Oil Crushing, Refining and Boiling *3 50 

Vol. II. Varnish Materials and Oil Varnish Making *4 00 

Vol. HI. Spirit Varnishes and Materials *4 5° 

McKnight, J. D., and Brown, A. W. Marine Multitubular Boilers *i 50 



IS D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

McMaster, J. B. Bridge and Tunnel Centres. (Science Series No. 20.) 

i6mo, o 50 

McMechen, F. L. Tests for Ores, Minerals and Metals nmo, *i 00 

McNeill, B. McNeill's Code 8vo, *6 00 

McPherson, J. A. Water- works Distribution 8vo, 2 50 

Melick, C. W. Dairy Laboratory Guide nmo, *i 25 

Merck, E. Chemical Reagents; Their Purity and Tests 8vo, *i 50 

Merritt, Wm. H. Field Testing for Gold and Silver . i6mo, leather, 1 50 

Messer, W. A. Railway Permanent Way 8vo, (In Press.) 

Meyer, J. G. A., and Pecker, C. G. Mechanical Drawing and Machine 

Design 4to, 5 00 

Michell, S. Mine Drainage 8vo, 10 00 

Mierzinski, S. Waterproofing of Fabrics. Trans, by A. Morris and H. 

Robson 8vo, *2 50 

Miller, E. H. Quantitative Analysis for Mining Engineers 8vo, *i 50 

Miller, G. A. Determinants. (Science Series No. 105.) i6mo, 

Milroy, M. E. W. Home Lace-making nmo, *i 00 

Minifie, W. Mechanical Drawing 8vo, *4 00 

Mitchell, C. A., and Prideaux, R. M. Fibres Used in Textile and Allied 

Industries 8vo, *3 00 

Modern Meteorology 121110, 1 50 

Monckton, C. C. F. Radiotelegraphy. (Westminster Series.) 8vo, *2 00 

Monteverde, R. D. Vest Pocket Glossary of English-Spanish, Spanish- 
English Technical Terms 64mo, leather, *i 00 

Moore, E. C. S. New Tables for the Complete Solution of Ganguillet and 

Kutter's Formula 8vo, *5 00 

Morecroft, J. H., and Hehre, F. W. Short Course in Electrical Testing. 

8vo, *i 50 

Moreing, C. A., and Neal, T. New General and Mining Telegraph Code, 8vo, *5 00 

Morgan, A. P. Wireless Telegraph Apparatus for Amateurs nmo, *i 50 

Moses, A. J. The Characters of Crystals 8vo, *2 00 

Moses, A. J., and Parsons, C. L. Elements of Mineralogy 8vo, *2 50 

Moss, S. A. Elements of Gas Engine Design. (Science Series No. 121. )i6mo, o 50 

The Lay-out of Corliss Valve Gears. (Science Series No. 119.). i6mo, o 50 

Mulford, A. C. Boundaries and Landmarks . -v- (In Press.) 

Mullin, J. P. Modern Moulding and Pattern-making i2mo, 2 50 

Munby, A. E. Chemistry and Physics of Building Materials. (Westmin- 
ster Series.) 8vo, *2 00 

Murphy, J. G. Practical Mining i6mo, 1 00 

Murray, J. A. Soils and Manures. (Westminster Series.) 8vo, *2 00 

Naquet, A. Legal Chemistry .' i2mo, 2 00 

Nasmith, J. The Student's Cotton Spinning 8vo, 3 r oo 

• Recent Cotton Mill Construction i2mo, 2" 00 

Neave, G. B., and Heilbron, I. M. Identification of Organic Compounds. h JN 

i2mo, *i 25 

Neilson, R. M. Aeroplane Patents 8vo, *2 00 

Nerz, F. Searchlights. Trans, by C. Rodgers 8vo, *3 00 

Nesbit, A. F. Electricity and Magnetism (In Preparation.) 

Neuberger, H., and Noalhat, H. Technology of Petroleum. Trans, by J. 

G. Mcintosh 8vo, *io 00 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 19 

Newall, J. W. Drawing, Sizing and Cutting Bevel-gears 8vo, i 50 

Nicol, G. Ship Construction and Calculations 8vo, *4 50 

Nipher, F. E. Theory of Magnetic Measurements i2mo, 1 00 

Nisbet, H. Grammar of Textile Design 8vo, *3 00 

Nolan, H. The Telescope. (Science Series No. 51.) i6mo, o 50 

Noll, A. How to Wire Buildings i2mo, 1 50 

North, H. B. Laboratory Notes of Experiments and General Chemistry. 

(In Press.) 

Nugent, E. Treatise on Optics nmo, 1 50 

O'Connor, H. The Gas Engineer's Pocketbook i2mo, leather, 3 50 

Petrol Air Gas i2mo, *o 75 

Ohm, G. S., and Lockwood, T. D. Galvanic Circuit. Translated by 

William Francis. (Science Series No. 102.) i6mo, o 50 

Olsen, J. C. Text-book of Quantitative Chemical Analysis 8vo, *4 00 

Olsson, A. Motor Control, in Turret Turning and Gun Elevating. (U. S. 

Navy Electrical Series, No. 1.) i2mo, paper, *o 50 

Oudin, M. A. Standard Polyphase Apparatus and Systems 8vo, *3 00 

Pakes, W. C. C, and Nankivell, A. T. The Science of Hygiene. .8vo, *i 75 

Palaz, A. Industrial Photometry. Trans, by G. W. Patterson, Jr. . . 8vo, *4 00 

Pamely, C. Colliery Manager's Handbook 8vo, *io 00 

Parr, G. D. A. Electrical Engineering Measuring Instruments 8vo, *s 5° 

Parry, E. J. Chemistry of Essential Oils and Artificial Perfumes. . . .8vo, *5 00 

Foods and Drugs. Two Volumes 8vo, 

Vol. I. Chemical and Microscopical Analysis of Foods and Drugs. *7 50 

Vol. H. Sale of Food and Drugs Act '. *3 00 

Parry, E. J., and Coste, J. H. Chemistry of Pigments 8vo, *4 50 

Parry, L. A. Risk and Dangers of Various Occupations 8vo, *3 00 

Parshall, H. F., and Hobart, H. M. Armature Windings 4to, *7 50 

Electric Railway Engineering 4to, *io 00 

Parshall, H. F., and Parry, E. Electrical Equipment of Tramways. . . . (In Press.) 

Parsons, S. J. Malleable Cast Iron 8vo, *2 50 

Partington, J. R. Higher Mathematics for Chemical Students. .i2mo, *2 00 

Passmore, A. C. Technical Terms Used in Architecture 8vo, *3 50 

Paterson, G. W. L. Wiring Calculations i2mo, *2 00 

Patterson, D. The Color Printing of Carpet Yarns 8vo, *3 50 

Color Matching on Textiles 8vo, *3 00 

The Science of Color Mixing 8vo, *3 00 

Paulding, C. P. Condensation of Steam in Covered and Bare Pipes. 

8vo, *2 00 

Transmission of Heat through Cold-storage Insulation nmo, *i 00 

Payne, D. W. Iron Founders' Handbook (In Press.) 

Peddie, R. A. Engineering and Metallurgical Books nmo, 

Peirce, B. System of Analytic Mechanics 4to, 10 00 

Pendred, V. The Railway Locomotive. (Westminster Series.) 8vo, *2 00 

Perkin, F. M. Practical Methods of Inorganic Chemistry nmo, *i 00 

Perrigo, 0. E. Change Gear Devices 8vo, 1 00 

Perrine, F. A. C. Conductors for Electrical Distribution 8vo, *3 50 

Perry, J. Applied Mechanics 8vo, *2 50 

Petit, G. White Lead and Zinc White Paints 8vo, *i 50 



20 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Petit, R. How to Build an Aeroplane. Trans, by T. O'B. Hubbard, and 

J. H. Ledeboer 8vo, 

Pettit, Lieut. J. S. Graphic Processes. (Science Series No. 76.) . . . i6mo, 
Philbrick, P. H. Beams and Girders. (Science Series No. 88.) . . . i6mo, 

Phillips, J. Engineering Chemistry 8vo, 

Gold Assaying 8vo, 

Dangerous Goods 8vo, 

Phin, J. Seven Follies of Science i2mo, 

Pickworth, C. N. The Indicator Handbook. Two Volumes. . i2mo, each, 

Logarithms for Beginners 121110, boards, 

The Slide Rule nmo, 

Plattner's Manual of Blow-pipe Analysis. Eighth Edition, revised. Trans. 

by H. B. Cornwall 8vo, 

Plympton, G. W. The Aneroid Barometer. (Science Series No. 35.) i6mo, 

How to become an Engineer. (Science Series No. 100.) . . i6mo, 

Van Nostrand's Table Book. (Science Series No. 104.) i6mo, 

Pochet, M. L. Steam Injectors. Translated from the French. (Science 

Series No. 29.) i6mo, 

Pocket Logarithms to Four Places. (Science Series No. 65.) i6mo, 

leather, 

Polleyn, F. Dressings and Finishings for Textile Fabrics 8vo, 

Pope, F. L. Modern Practice of the Electric Telegraph 8vo, 

Popplewell, W. C. Elementary Treatise on Heat and Heat Engines. . i2mo, 

Prevention of Smoke 8vo, 

Strength of Materials 8vo, 

Porter, J. R. Helicopter Flying Machine i2mo, 

Potter, T. Concrete 8vo, 

Potts, H. E. Chemistry of the Rubber Industry. (Outlines of Indus- 
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Practical Iron Founding i2mo, 

Pratt, K. Boiler Draught nmo, 

Pray, T., Jr. Twenty Years with the Indicator 8vo, 

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Calorimeter Tables 8vo, 

Preece, W. H. Electric Lamps {In Press.) 

Prelini, C. Earth and Rock Excavation 8vo, 

Graphical Determination of Earth Slopes 8vo, 

Tunneling. New Edition 8vo, 

Dredging. A Practical Treatise 8vo, 

Prescott, A. B. Organic Analysis 8vo, 

Prescott, A. B., and Johnson, 0. C. Qualitative Chemical Analysis. . .8vo, 
Prescott, A. B., and Sullivan, E. C. First Book in Qualitative Chemistry. 

i2mo, 

Prideaux, E. B. R. Problems in Physical Chemistry 8vo, 

Pritchard, 0. G. The Manufacture of Electric-light Carbons . . 8vo, paper, 
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Structures i2mo, 

Injectors: Theory, Construction and Working nmo, 

Pulsifer, W. H. Notes for a History of Lead 8vo, 

Purchase, W. R. Masonry i2mo, 



*I 


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00 


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50 


1 


50 


*i 


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2 


00 


1 


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*3 


00 


*2 


00 


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00 


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00 


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50 


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50 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 21 

Putsch, A. Gas and Coal-dust Firing 8vo, *3 00 

Pynchon, T. R. Introduction to Chemical Physics 8vo, 3 00 

Rafter G. W, Mechanics of Ventilation. (Science Series No. 33.) . i6mo, 

Potable Water, (Science Series No. 103.) i6mc 

Treatment of Septic Sewage. (Science Series No. 118.). . . . i6mo 

Rafter, G. W., and Baker, M. N. Sewage Disposal in the United States. 

4to, 

Raikes, H. P. Sewage Disposal Works 8vo, 

Railway Shop Up-to-Date 4to, 

Ramp, H. M. Foundry Practice (In Press.) 

Randall, P. M. Quartz Operator's Handbook nmo, 

Randau, P. Enamels and Enamelling 8vo, 

Rankine, W. J. M. Applied Mechanics '. 8vo, 

Civil Engineering 8vo, 

Machinery and Millwork 8vo, 

The Steam-engine and Other Prime Movers 8vo, 

Useful Rules and Tables 8vo, 

Rankine, W. J. M., and Bamber, E. F. A Mechanical Text-book. . . .8vo, 
Raphael, F. C. Localization of Faults in Electric Light and Power Mains. 

8vo, *3 00 

Rasch, E. Electric Arc. Trans, by K. Tornberg (In Press.) 

Rathbone, R. L. B. Simple Jewellery 8vo, *2 00 

Rateau, A. Flow of Steam through Nozzles and Orifices. Trans, by H. 

B. Brydon 8vo, *i 50 

Rausenberger, F. The Theory of the Recoil of Guns 8vo, *4 50 

Rautenstrauch, W. Notes on the Elements of Machine Design. 8vo, boards, *i 50 
Rautenstrauch, W., and Williams, J. T. Machine Drafting and Empirical 

Design. 

Part I. Machine Drafting 8vo, *i 25 

Part II. Empirical Design (In Preparation.) 

Raymond, E. B. Alternating Current Engineering nmo, *2 50 

Rayner, H. Silk Throwing and Waste Silk Spinning 8vo, *2 50 

Recipes for the Color, Paint, Varnish, Oil, Soap and Drysaltery Trades . 8vo, *3 50 

Recipes for Flint Glass Making nmo, *4 50 

Redfern, J. B. Bells, Telephones (Installation Manuals Series) i6mo, 

(In Press.) 

Redwood, B. Petroleum. (Science Series No. 92.) i6mo, o 50 

Reed's Engineers' Handbook 8vo, *5 00 

Key to the Nineteenth Edition of Reed's Engineers' Handbook. .8vo, *3 00 

Useful Hints to Sea-going Engineers nmo, 1 50 

Marine Boilers nmo, 2 00 

Reinhardt, C. W. Lettering for Draftsmen, Engineers, and Students. 

oblong 4to, boards, 1 00 

The Technic of Mechanical Drafting oblong 4to, boards, *i 00 

Reiser, F. Hardening and Tempering of Steel. Trans, by A. Morris and 

H. Robson nmo, *2 50 

Reiser, N. Faults in the Manufacture of Woolen Goods. Trans, by A. 

Morris and H. Robson 8vo, *2 50 

Spinning and Weaving Calculations 8vo, *5 00 

Renwick, W. G. Marble and Marble Working 8vo, 5 00 



22 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Reynolds, 0., and Idell, F. E. Triple Expansion Engines. (Science 

Series No. 99.) i6mo, o 50 

Rhead, G. F. Simple Structural Woodwork i2mo, *i 00 

Rice, J. M., and Johnson, W. W. A New Method of Obtaining the Differ- 
ential of Functions nmo, o 50 

Richards, W. A. and North, H. B. Manual of Cement Testing. (In Press.) 

Richardson, J. The Modern Steam Engine 8vo, 

Richardson, S. S. Magnetism and Electricity i2mo, 

Rideal, S. Glue and Glue Testing 8vo, 

Rings, F. Concrete in Theory and Practice i2mo, 

Ripper, W. Course of Instruction in Machine Drawing folio, 

Roberts, F. C. Figure of the Earth. (Science Series No. 79.) i6mo, 

Roberts, J., Jr. Laboratory Work in Electrical Engineering 8vo, 

Robertson, L. S. Water-tube Boilers 8vo, 

Robinson, J. B. Architectural Composition 8vo, 

Robinson, S. W. Practical Treatise on the Teeth of Wheels. (Science 

Series No. 24.) i6mo, 

Railroad Economics. (Science Series No. 59.) i6mo, 

Wrought Iron Bridge Members. (Science Series No. 60.) i6mo, 

Robson, J. H. Machine Drawing and Sketching 8vo, 

Boebling, J A. Long and Short Span Railway Bridges folio, 

Rogers, A. A Laboratory Guide of Industrial Chemistry nmo, 

Rogers, A., and Aubert, A. B. Industrial Chemistry 8vo, 

Rogers, F, Magnetism of Iron Vessels. (Science Series No. 30.) . . i6mo, 
Rohland, P. Colloidal and Cyrstalloidal State of Matter. Trans, by 

W. J. Britland and H. E. Potts i2mo, 

Rollins, W. Notes on X-Light 8vo, 

Rollinson, C. Alphabets Oblong, .i2mo, (In Press.) 

Rose, J. The Pattern-makers' Assistant 8vo, 

Key to Engines and Engine-running nmo, 

Rose, T. K. The Precious Metals. (Westminster Series.) 8vo, 

Rosenhain, W. Glass Manufacture. (Westminster Series.) 8vo, 

Ross, W. A. Plowpipe in Chemistry and Metallurgy nmo, 

Rossiter, J. T. Steam Engines. (Westminster Series.). . . .8vo (In Press.) 

Pumps and Pumping Machinery. (Westminster Series.).. 8vo (In Press.) 

Roth. Physical Chemistry 8vo, 

Rouillion, L. The Economics of Manual Training 8vo, 

Rowan, F. J. Practical Physics of the Modern Steam-boiler 8vo, 

Rowan, F. J., and Idell, F. E. Boiler Incrustation and Corrosion. 

(Science Series No. 27.) i6mo, 

Roxburgh, W. General Foundry Practice 8vo, 

Ruhmer, E. Wireless Telephony. Trans, by J. Erskine-Murray. . . .8vo, 
Russell, A. Theory of Electric Cables and Networks 8vo, 

Sabine, R. History and Progress of the Electric Telegraph nmo, 

Saeltzer A. Treatise on Acoustics nmo, 

Salomons, D. Electric Light Installations, nmo. 

Vol. I. The Management of Accumulators 

Vol. II. Apparatus 

Vol. III. Applications 

Sanford, P. G. Nitro-explosives 8vo, 



*3 


50 


*2 


00 


*4 


00 


*2 


50 


*6 


00 





50 


*2 


00 


3 


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50 





50 





50 


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50 


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50 


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D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 23 

Saunders, C. H. Handbook of Practical Mechanics i6mo, 

leather, 

Saunnier, C. Watchmaker's Handbook nmo, 

Sayers, H. M. Brakes for Tram Cars 8vo, 

Scheele, C. W. Chemical Essays 8vo, 

Schellen, H. Magneto-electric and Dynamo-electric Machines 8vo, 

Scherer, R. Casein. Trans, by C. Salter 8vo, 

Schidrowitz, P. Rubber, Its Production and Industrial Uses 8vo, 

Schindler, K. Iron and Steel Construction Works. 

Schmall, C. N. First Course in Analytic Geometry, Plane and Solid. 

i2mo, half leather, 
Schmall, C. N., and Shack, S. M. Elements of Plane Geometry .... i2mo, 

Schmeer, L. Flow of Water 8vo, 

Schumann, F. A Manual of Heating and Ventilation i2mo, leather, 

Schwarz, E. H. L. Causal Geology 8vo, 

Schweizer, V., Distillation of Resins 8vo, 

Scott, W. W. Qualitative Analysis. A Laboratory Manual 8vo, 

Scribner, J. M. Engineers' and Mechanics' Companion . . . i6mo, leather, 

Searle, A. B. Modern Brickmaking 8vo, 

Searle, G. M. " Sumners' Method." Condensed and Improved. (Science 

Series No. 124.) i6mo, 

Seaton, A. E. Manual of Marine Engineering 8vo, 

Seaton, A. E., and Rounthwaite, H. M. Pocket-book of Marine Engineer- 
ing i6mo, leather, 3 00 

Seeligmann, T., Torrilhon, G. L., and Falconnet, H. India Rubber and 

Gutta Percha. Trans, by J. G. Mcintosh 8vo, 

Seidell, A. Solubilities of Inorganic and Organic Substances 8vo, 

Sellew, W. H. Steel Rails ^to {In Press.) 

Senter, G. Outlines of Physical Chemistry nmo, 

Textbook of Inorganic Chemistry i2mo, 

Sever, G. F. Electric Engineering Experiments 8vo, boards, 

Sever, G. F., and Townsend, F. Laboratory and Factory Tests in Electrical, 

Engineering 8vo, 

Sewall, C. H. Wireless Telegraphy 8vo, 

Lessons in Telegraphy i2mo, 

Sewell, T. Elements of Electrical Engineering 8vo, 

The Construction of Dynamos 8mo, 

Sexton, A. H. Fuel and Refractory Materials nmo, 

• Chemistry of the Materials of Engineering nmo, 

Alloys (Non-Ferrous) 8vo, 

The Metallurgy of Iron and Steel 8vo, 

Seymour, A. Practical Lithography 8vo, 

Modern Printing Inks 8vo, 

Shaw, Henry S. H. Mechanical Integrators. (Science Series No. 83.) 

i6mo, 

Shaw, P. E. Course of Practical Magnetism and Electricity 8vo, 

Shaw, S. History of the Staffordshire Potteries 8vo, 

Chemistry of Compounds Used in Porcelain Manufacture 8vo, 

Shaw, W. N. Forecasting Weather 8vo, 

Sheldon, S., and Hausmann, E. Direct Current Machines nmo, 

Alternating Current Machines i2mo, 



*5 


00 


*3 


00 


*i 


75 


*i 


75 


*i 


00 


*2 


50 


*2 


00 


*I 


00 


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00 


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00 


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*6 


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50 


*I 


00 


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00 


*3 


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*2 


50 


*2 


50 



24 D. VAX NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Sheldon, S., and Hausmann, E. Electric Traction and Transmission 

Engineering i2mo, 

Sherriff, F. F. Oil Merchants' Manual nmo, 

Shields, J. E. Notes on Engineering Construction , i2mo, 

Shock, W. H. Steam Boilers 4to, half morocco, 

Shreve, S. H. Strength of Bridges and Roofs 8vo, 

Shunk, W. F. The Field Engineer nmo, morocco, 

Simmons, W. H., and Appleton, H. A. Handbook of Soap Manufacture. 

8vo, 

Simmons, W. H., and Mitchell, C. A. Edible Fats and Oils 8vo, 

Simms, F. W. The Principles and Practice of Leveling 8vo, 

Practical Tunneling 8vo, 

Simpson, G. The Naval Constructor i2mo, morocco, 

Simpson, W. Foundations 8vo, (In Press.) 

Sinclair, A. Development of the Locomotive Engine . . . 8vo, half leather, 

Sinclair, A. Twentieth Century Locomotive 8vo, half leather, 

Sindall, R. W. Manufacture of Paper. (Westminster Series.) 8vo, 

Sloane, T. O'C. Elementary Electrical Calculations nmo, 

Smith, C. A. M. Handbook of Testing, MATERIALS 8vo, 

Smith, C. A. M., and Warren, A. G. New Steam Tables 8vo, 

Smith, C. F. Practical Alternating Currents and Testing 8vo, 

Practical Testing of Dynamos and Motors 8vo, 

Smith, F. E. Handbook of General Instruction for Mechanics. . . . nmo, 

Smith, J. C. Manufacture of Paint 8vo, 

Smith, R. H. Principles of Machine Work i2mo, 

Elements of Machine Work i2mo, 

Smith, W. Chemistry of Hat Manufacturing nmo, 

Snell, A. T. Electric Motive Power 8vo, 

Snow, W. G. Pocketbook of Steam Heating and Ventilation. (In Press.) 
Snow, W. G., and Nolan, T. Ventilation of Buildings. (Science Series 

No. 5.) i6mo, 

Soddy, F. Radioactivity 8vo, 

Solomon, M. Electric Lamps. (Westminster Series.) 8vo, 

Sothern, J. W. The Marine Steam Turbine 8vo, 

Southcombe, J. E. Paints, Oils and Varnishes. (Outlines of Indus- 
trial Chemistry.) 8vo, (In Press.) 

Soxhlet, D. H. Dyeing and Staining Marble. Trans, by A. Morris and 

H. Robson 8vo, *2 50 

Spang, H. W. A Practical Treatise on Lightning Protection nmo, 1 00 

Spangenburg, L. Fatigue of Metals. Translated by S. H. Shreve. 

(Science Series No. 23.) i6mo, o 50 

Specht, G. J., Hardy, A. S., McMaster, J.B ., and Walling. Topographical 

Surveying. (Science Series No. 72.) i6mo, 

Speyers, C. L. Text-book of Physical Chemistry 8vo, 

Stahl, A. W. Transmission of Power. (Science Series No. 28.) . . . i6mo, 

Stahl, A. W., and Woods, A. T. Elementary Mechanism nmo, 

Staley, C, and Pierson, G. S. The Separate System of Sewerage. . . .8vo, 

Standage, H. C. Leatherworkers' Manual 8vo, 

Sealing Waxes, Wafers, and Other Adhesives 8vo, 

Agglutinants of all Kinds for all Purposes nmo, 

Stansbie, J. H. Iron and Steel. (Westminster Series.) 8vo, 



*2 


50 


*3 


5o 


I 


50 


15 


00 


3 


50 


2 


50 


*3 


00 


*3 


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2 


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7 


50 


*5 


00 


5 


00 


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D. VAX XOSTRAXD COMPANY'S SHORT TITLE CATALOG 25 

Steinman, D. B. Suspension Bridges and Cantilevers. (Science Series 

No. 127) o 50 

Stevens, H. P. Paper Mill Chemist i6mo, *2 50 

Stevenson, J. L. Blast-Furnace Calculations i2mo, leather, *2 00 

Stewart, A. Modern Polyphase Machinery i2mo, *2 00 

Stewart, G. Modern Steam Traps i2mo, *i 25 

Stiles, A. Tables for Field Engineers i2mo, 1 00 

Stiflman, P. Steam-engine Indicator i2mo, 1 00 

Stodola, A. Steam Turbines. Trans, by L. C. Loewenstein 8vo, *5 00 

Stone, H. The Timbers of Commerce 8vo, 3 50 

Stone, Gen. R. New Roads and Road Laws nmo, 1 00 

Stopes, M. Ancient Plants 8vo, *2 00 

The Study of Plant Life 8vo, *2 00 

Stumpf, Prof. Una-Flow of Steam Engine (In Press.) 

Sudborough, J. J., and James, T. C. Practical Organic Chemistry. . nmo, *2 00 

Suffling, E. R. Treatise on the Art of Glass Painting 8vo, *3 50 

Swan, K. Patents, Designs and Trade Marks. (Westminster Series. ).8vo, *2 00 

Sweet, S. H. Special Report on Coal 8vo, 3 00 

Swinburne, J., Wordingham, C. H., and Martin, T. C. Eletcric Currents. 

(Science Series No. 109.) i6mo, o 50 

Swoope, C. W. Practical Lessons in Electricity i2mo, *2 00 

Tailfer, L. Bleaching Linen and Cotton Yarn and Fabrics 8vo, *5 00 

Tate, J. S. Surcharged and Different Forms of Retaining-walls. (Science 

Series No. 7.) i6mo, 50 

Taylor, E. N. Small Water Supplies nmo, *2 00 

Templeton, W. Practical Mechanic's Workshop Companion. 

nmo, morocco, 2 00 
Terry, H. L. India Rubber and its Manufacture. (Westminster Series.) 

8vo, *2 00 
Thayer, H. R. Structural Design. 8vo. 

Vol. I. Elements of Structural Design *2 00 

Vol. II. Design of Simple Structures (In Preparation.) 

Vol. HI. Design of Advanced Structures (In Preparation.) 

Thiess, J. B. and Joy, G. A. Toll Telephone Practice . 8vo, *3 50 

Thorn, C, and Jones, W. H. Telegraphic Connections oblong nmo, 1 50 

Thomas, C. W. Paper-makers' Handbook (In Press.) 

Thompson, A. B. Oil Fields of Russia 4to, *7 50 

Petroleum Mining and Oil Field Development 8vo, *5 00 

Thompson, E. P. How to Make Inventions 8vo, 50 

Thompson, S. P. Dynamo Electric Machines. (Science Series No. 75.) 

i6mo, 50 

Thompson, W. P. Handbook of Patent Law of All Countries i6mo, 1 50 

Thomson, G. S. Milk and Cream Testing nmo, *i 75 

Modern Sanitary Engineering, House Drainage, etc. 8vo, (In Press.) 

Thornley, T. Cotton Combing Machines 8vo, *3 00 

Cotton Spinning. 8vo. 

First Year *i 50 

Second Year *2 50 

Third Year *2 50 

Thurso, J. W. Modern Turbine Practice 8vo, *4 00 



o 


50 


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00 


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oo 


*3 


00 


*2 


00 


*7 


50 


*2 


00 


*o 


75 


*I 


25 



26 D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Tidy, C. Meymott. Treatment of Sewage. (Science Series No. 94.). 

i6mo, 

Tinney, W. H. Gold-mining Machinery 8vo, 

Titherley, A. W. Laboratory Course of Organic Chemistry 8vo, 

Toch, M. Chemistry and Technology of Mixed Paints 8vo, 

Materials for Permanent Painting i2mo, 

Todd, J., and Whall, W. B. Practical Seamanship 8vo, 

Tonge, J. Coal. (Westminster Series.) 8vo, 

Townsend, F. Alternating Current Engineering 8vo, boards 

Townsend, J. Ionization of Gases by Collision 8vo, 

Transactions of the American Institute of Chemical Engineers. 8vo. 

Vol. I. 1908 *6 00 

Vol. II. 1909 *6 00 

Vol. III. 1910 *6 00 

Vol. IV. 191 1 : *6 00 

Traverse Tables. (Science Series No. 115.) i6mo, 

morocco, 
Trinks, W., and Housum, C. Shaft Governors. (Science Series No. 122.) 

i6mo, 
Trowbridge, W. P. Turbine Wheels. (Science Series No. 44.) . . : . . i6mo, 

Tucker, J. H. A Manual of Sugar Analysis 8vo, 

Tumlirz, 0. Potential. Trans, by D. Robertson i2mo, 

Tunner, P. A. Treatise on Roll-turning. Trans, by J. B. Pearse. 

8vo, text and folio atlas, 

Turbayne, A. A. Alphabets and Numerals 4to, 

Turnbull, Jr., J., and Robinson, S. W. A Treatise on the Compound 

Steam-engine, (Science Series No. 8.) i6mo, 

Turrill, S. M. Elementary Course in Perspective i2mo, *i 25 

Underhill, C. R. Solenoids, Electromagnets and Electromagnetic Wind- 
ings i2mo, *2 00 

Universal Telegraph Cipher Code nmo, 1 00 

Urquhart, J. W. Electric Light Fitting i2mo, 2 00 

Electro-plating i2mo, 2 00 

Electrotyping i2mo, 2 00 

Electric Ship Lighting nmo, 3 00 

Vacher, F. Food Inspector's Handbook nmo, *2 50 

Van Nostrand's Chemical Annual. Second issue 1909 nmo, *2 50 

Year Book of Mechanical Engineering Data. First issue 1912 . . . (In Press.) 

Van Wagenen, T. F. Manual of Hydraulic Mining i6mo, 1 00 

Vega, Baron Von. Logarithmic Tables 8vo, half morocco, 2 00 

Villon, A. M. Practical Treatise on the Leather Industry. Trans, by F. 

T. Addyman 8vo, *io 00 

Vincent, C. Ammonia and its Compounds. Trans, by M. J. Salter . . 8vo, *2 00 

Volk, C. Haulage and Winding Appliances 8vo, *4 00 

Von Georgievics, G. Chemical Technology of Textile Fibres. Trans, by 

C. Salter 8vo, *4 50 

Chemistry of Dyestuffs. Trans, by C. Salter 8vo, *4 50 

Vose, G. L. Graphic Method for Solving Certain Questions in Arithmetic 

and Algebra. (Science Series No. 16.) i6mo, o 50 






50 


I 


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50 





50 


3 


50 


1 


25 


10 


00 


2 


00 



D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 27 

Wabner, R. Ventilation in Mines. Trans, by C. Salter 8vo, *4 50 

Wade, E. J. Secondary Batteries 8vo, *4 00 

Wadmore, T. M. Elementary Chemical Theory.. nmo, *i 50 

Wadsworth, C. Primary Battery Ignition nmo (In Press.) 

Wagner, E. Preserving Fruits, Vegetables, and Meat i2mo, *2 50 

Waldram, P. J. Principles of Structural Mechanics (In Press.) 

Walker, F. Aerial Navigation 8vo, 

Dynamo Building. (Science Series No. 98.) i6mo, 

Electric Lighting for Marine Engineers 8vo, 

Walker, S. F. Steam Boilers, Engines and Turbines 8vo, 

Refrigeration, Heating and Ventilation on Shipboard i2mo, 

Electricity in Mining 8vo, 

Walker, W. H. Screw Propulsion 8vo, 

Wallis-Tayler, A. J. Bearings and Lubrication 8vo, 

Aerial or Wire Ropeways 8vo, 

Modern Cycles 8vo, 

Motor Cars 8vo, 

Motor Vehicles for Business Purposes 8vo, 

Pocket Book of Refrigeration and Ice Making nmo, 

Refrigeration, Cold Storage and Ice-Making 8vo, 

Sugar Machinery nmo, 

Wanklyn, J. A. Water Analysis nmo, 

Wansbrough, W. D. The A B C of the Differential Calculus nmo, 

Slide Valves nmo, 

Ward, J. H. Steam for the Million 8vo, 

Waring, Jr., G. E. Sanitary Conditions. (Science Series No. 31.)- • i6mo, 

Sewerage and Land Drainage *6 

Waring, Jr., G. E. Modern Methods of Sewage Disposal nmo, 

How to Drain a House nmo, 

Warren, F. D. Handbook on Reinforced Concrete nmo, 

Watkins, A. Photography. (Westminster Series.) 8vo, 

Watson, E. P. Small Engines and Boilers nmo, 

Watt, A. Electro-plating and Electro-refining of Metals 8vo, 

Electro-metallurgy nmo, 

The Art of Soap-making 8vo, 

Leather Manufacture 8vo, 

Paper-Making 8vo, 

Weale, J. Dictionary of Terms Used in Architecture nmo, 

Weale's Scientific and Technical Series. (Complete list sent on applica- 
tion.) 
Weather and Weather Instruments nmo, 

paper, 
Webb, H. L. Guide to the Testing of Insulated Wires and Cables. . nmo, 

Webber, W. H. Y. Town Gas. (Westminster Series.) 8vo, 

Weisbach, J. A Manual of Theoretical Mechanics 8vo, 

sheep, 

Weisbach, J., and Herrmann, G. Mechanics of Air Machinery 8vo, 

Welch, W. Correct Lettering (In Press.) 

Weston, E. B. Loss of Head Due to Friction of Water in Pipes . . . nmo, 

Weymouth, F. M. Drum Armatures and Commutators 8vo, 

Wheatley, O. Ornamental Cement Work (In Press.) 



2 


00 





50 


2 


00 


3 


00 


*2 


00 


*3 


50 





75 


*I 


50 


*3 


00 


4 


00 


1 


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1 


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50 


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50 



2S D. VAN NOSTRAND COMPANY'S SHORT TITLE CATALOG 

Wheeler, J. B. Art of War nmo, i 75 

Field Fortifications i2mo, 1 75 

Whipple, S. An Elementary and Practical Treatise on Bridge Building. 

8vo, 3 00 

Whithard, P. Illuminating and Missal Painting i2mo, 1 50 

Wilcox, R. M. Cantilever Bridges. (Science Series No. 25.) i6mo, o 50 

Wilkinson, H. D. Submarine Cable Laying and Repairing 8vo, *6 00 

Williams, A. D., Jr., and Hutchinson, R. W. The Steam Turbine (In Press.) 

Williamson, J., and Blackadder, H. Surveying 8vo, (In Press.) 

Williamson, R. S. On the Use of the Barometer 4to, 

Practical Tables in Meteorology and Hypsometery 4to, 

Willson, F. N. Theoretical and Practical Graphics 4T0, 

Wimperis, H. E. Internal Combustion Engine 8vo, 

Winchell, H. H., and A. N. Elements of Optical Mineralogy 8vo, 

Winkler, C, and Lunge, G. Handbook of Technical Gas- Analysis . . .8vo, 

Winslow, A. Stadia Surveying. (Science Series No. 77.) i6mo, 

Wisser, Lieut. J. P. Explosive Materials. (Science Series No. 70.). 

i6mo, 
Wisser, Lieut. J. P. Modern Gun Cotton. (Science Series No. 89.) i6mo, 
Wood, De V. Luminiferous Aether. (Science Series No. 85.) .... i6mo, 
Woodbury, D. V. Elements of Stability in the Well-proportioned Arch. 

8vo, half morocco, 4 00 

Worden, E. C. The Nitrocellulose Industry. Two Volumes 8vo, *io 00 

Cellulose Acetate 8vo, (In Press.) 

Wright, A. C. Analysis of Oils and Allied Substances 8vo, 

■ Simple Method for Testing Painters' Materials 8vo, 

Wright, F. W. Design of a Condensing Plant nmo, 

Wright, H. E. Handy Book for Brewers 8vo, 

Wright, J. Testing, Fault Finding, etc., for Wiremen. (Installation 

Manuals Series.) i6mo, 

Wright, T. W. Elements of Mechanics 8vo, 

Wright, T. W., and Hayford, J. F. Adjustment of Observations.. . . .8vo, 

Young, J. E. Electrical Testing for Telegraph Engineers 8vo, *4 00 

Zahner, R. Transmission of Power. (Science Series No. 40.) .... i6mo, 

Zeidler, J., and Lustgarten, J. Electric Arc Lamps 8vo, *2 00 

Zeuner, A. Technical Thermodynamics. Trans, by J. F. Klein. Two 

Volumes 8vo, *8 00 

Zimmer, G. F. Mechanical Handling of Material 4to, *io 00 

Zipser, J. Textile Raw Materials. Trans, by C. Salter 8vo, *5 00 

Zur Nedden, F. Engineering Workshop Machines and Processes. Trans. 

by J. A. Davenport 8vo *2 00 



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